02.12.13 · analysis / ode

Inhomogeneous linear ODE / variation of constants

shipped3 tiersLean: none

Anchor (Master): Arnold *Ordinary Differential Equations* Ch.3 §§15-19 and Ch.4 §§24-29 (3rd ed., Springer 1992); Hartman *Ordinary Differential Equations* (2nd ed., SIAM 2002) Ch.IV (linear systems on Banach spaces, Floquet theory, exponential dichotomies); Coddington-Levinson *Theory of Ordinary Differential Equations* (McGraw-Hill 1955) Ch.3 (the classical variation-of-constants framework); Lagrange 1808-1809 *Mécanique Analytique* second edition (V$\hat{\text{v}}$e Courcier) — variation of arbitrary constants in the perturbed Keplerian problem; Cauchy 1820s École Polytechnique lectures (Œuvres complètes Série II Tomes XI-XII) — existence theorem in the linear case via the polygon method; Liouville 1838 *Note sur la théorie de la variation des constantes arbitraires* (J. Math. Pures Appl. (1) 3, 342-349) — the determinant formula $\det \Phi(t) = \det \Phi(t_0) \exp \int \mathrm{tr}\, A(s)\, ds$; Duhamel 1830s *Mémoire sur la méthode générale relative au mouvement de la chaleur dans les corps solides plongés dans des milieux dont la température varie avec le temps* (J. École Polytechnique 14, 22e cahier, 1833, pp. 20-77) — convolution formula for the inhomogeneous heat / wave problem; Picard 1893 *Sur l'application des méthodes d'approximations successives à l'étude de certaines équations différentielles ordinaires* (J. Math. Pures Appl. (4) 9, 217-271) — successive-approximation construction of the fundamental matrix and the general linear-existence theorem

Intuition [Beginner]

Imagine you have a swing that, left alone, oscillates back and forth with a known rhythm — that is the homogeneous part of the story, the swing's natural motion. Now imagine someone pushes the swing periodically. The pushes are an external forcing, and the swing's motion is no longer just its natural oscillation: it now responds to the input as well. A linear ODE with a forcing term captures exactly this picture. The unforced equation has solutions that flow according to an internal rule; the forced equation adds an inhomogeneous source that the system must accommodate on top of its natural flow.

The variation-of-constants method is a recipe for solving the forced equation by reusing the solution to the unforced one. Lagrange's insight in 1809 was that if the homogeneous problem is already solved — meaning you know how the unforced system propagates initial conditions forward in time — then the forced problem can be solved by allowing the "constants of integration" to drift with time. The drift rate is determined by how much forcing is applied at each instant. The total response of the forced system is the homogeneous propagation of the initial condition plus an accumulated integral of past forcings weighted by the propagator. This is the principle of superposition stated with arithmetic precision.

Why does it matter? Almost every physical system that is small enough to be treated as linear — circuits, springs, mechanical filters, beams, acoustic waveguides — obeys a linear ODE with an inhomogeneous forcing term. Knowing how to combine the natural response with the forced response is the basic computational tool of engineering and applied physics. The same recipe extends, almost without modification, to linear partial differential equations via Duhamel's principle.

Visual [Beginner]

A schematic shows two curves on the same set of axes. The first curve, drawn in solid line, is the natural oscillation of an unforced damped oscillator: it decays toward zero from some initial displacement. The second curve, drawn in dashed line, is the forced response of the same oscillator under a sinusoidal driving force: it settles into a steady oscillation whose amplitude is controlled by the driving frequency. Beside the curves, a small inset shows the magnitude of the steady-state response as a function of the driving frequency, with a peak at resonance.

The picture conveys the structural content: a forced linear system has a response that is the homogeneous propagation of any initial state, plus an integral of the forcing convolved with the homogeneous propagator. The two pieces add. Each forcing instant contributes its own future ripple, and the ripples superpose because the equation is linear.

Worked example [Beginner]

Solve the scalar linear ODE $\overset{x}{˙} = - x + e^{- 2 t}$ on the real line with initial condition $x (0) = 1$ .

Step 1. Solve the homogeneous part. The unforced equation $\overset{y}{˙} = - y$ has solutions $y (t) = c e^{- t}$ for any constant $c$ . The propagator is $Φ (t) = e^{- t}$ , the function that sends an initial condition $y (0) = c$ to its value at time $t$ .

Step 2. Apply variation of constants. Look for a solution of the forced equation in the form $x (t) = c (t) \cdot e^{- t}$ , with the constant $c$ now allowed to vary in time. Differentiate to get $\overset{x}{˙} = \overset{c}{˙} e^{- t} - c e^{- t}$ . Substitute into the equation $\overset{x}{˙} = - x + e^{- 2 t}$ to obtain $\overset{c}{˙} e^{- t} - c e^{- t} = - c e^{- t} + e^{- 2 t}$ . The $- c e^{- t}$ terms cancel, leaving $\overset{c}{˙} e^{- t} = e^{- 2 t}$ , equivalently $\overset{c}{˙} = e^{- t}$ .

Step 3. Antidifferentiate $\overset{c}{˙} = e^{- t}$ with initial condition $c (0) = x (0) / e^{0} = 1$ . The antiderivative of $e^{- t}$ is $- e^{- t}$ , so accumulating from time $0$ to time $t$ gives $c (t) = c (0) + (1 - e^{- t}) = 1 + (1 - e^{- t}) = 2 - e^{- t}$ .

Step 4. Reassemble the answer. The solution is $x (t) = c (t) e^{- t} = (2 - e^{- t}) e^{- t} = 2 e^{- t} - e^{- 2 t}$ .

Step 5. Check the answer. At $t = 0$ : $x (0) = 2 - 1 = 1$ , matching the initial condition. The derivative is $\overset{x}{˙} = - 2 e^{- t} + 2 e^{- 2 t}$ , while $- x + e^{- 2 t} = - (2 e^{- t} - e^{- 2 t}) + e^{- 2 t} = - 2 e^{- t} + 2 e^{- 2 t}$ . The two agree, so the answer is correct.

What this tells us: the forced solution is the homogeneous propagator $e^{- t}$ times a function $c (t)$ that records how much the system has been pushed off the unforced trajectory by the source $e^{- 2 t}$ . The "constant of integration" $c$ varies in time, and its variation rate is exactly the forcing divided by the propagator. The same recipe applies in higher dimensions with a matrix-valued propagator.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Let $I \subseteq R$ be an open interval, $n \geq 1$ , and $A : I \to Mat_{n} (R)$ a continuous matrix-valued function. The homogeneous linear system with coefficient matrix $A$ is the ordinary differential equation $$ \dot x = A(t) x, \qquad x : I \to \mathbb{R}^n. $$ A continuous $b : I \to R^{n}$ defines the corresponding inhomogeneous linear system $$ \dot x = A(t) x + b(t). $$

Given an initial time $t_{0} \in I$ , the fundamental matrix of the homogeneous system normalised at $t_{0}$ is the unique solution $Φ : I \to Mat_{n} (R)$ of the matrix-valued initial-value problem $$ \dot \Phi(t) = A(t) , \Phi(t), \qquad \Phi(t_0) = I_n, $$ where each column of $Φ$ is itself a solution of the homogeneous vector equation, and $I_{n}$ is the $n \times n$ identity matrix. Existence and uniqueness on all of $I$ follows from the Picard-Lindelöf theorem applied componentwise to the matrix equation, using that $A$ is continuous and the right-hand side is linear in $Φ$ , hence globally Lipschitz on bounded intervals.

The columns of $Φ$ are $n$ linearly independent solutions of the homogeneous system. Conversely, any matrix-valued function $Ψ : I \to Mat_{n} (R)$ whose columns are $n$ linearly independent solutions is called a fundamental matrix solution (not normalised at $t_{0}$ ); the normalised one is $Φ (t) = Ψ (t) Ψ (t_{0})^{- 1}$ .

The Wronskian determinant is $W (t) = det Φ (t)$ , equivalently the determinant of any matrix whose columns are $n$ solutions. The Wronskian is non-vanishing wherever the columns are linearly independent and vanishes identically where they are dependent. Liouville's formula (proved below) gives the explicit time dependence $$ W(t) = W(t_0) \exp \left( \int_{t_0}^t \mathrm{tr}, A(s) , ds \right), $$ showing that the Wronskian either vanishes identically or never vanishes.

The variation-of-constants formula (Lagrange 1808-1809) gives the unique solution of the inhomogeneous initial-value problem $\overset{x}{˙} = A (t) x + b (t)$ , $x (t_{0}) = x_{0}$ as $$ x(t) = \Phi(t) \left( x_0 + \int_{t_0}^t \Phi(s)^{-1} b(s) , ds \right) = \Phi(t) x_0 + \int_{t_0}^t \Phi(t) \Phi(s)^{-1} b(s) , ds. $$ The matrix $Φ (t) Φ (s)^{- 1}$ is the propagator from time $s$ to time $t$ : it sends the value of a homogeneous solution at time $s$ to its value at time $t$ .

In the constant-coefficient case $A (t) = A$ independent of time, the fundamental matrix is the matrix exponential $$ \Phi(t) = e^{(t - t_0) A} = \sum_{k=0}^\infty \frac{(t - t_0)^k}{k!} A^k, $$ and the propagator $Φ (t) Φ (s)^{- 1} = e^{(t - s) A}$ depends only on the time difference $t - s$ . The variation-of-constants formula specialises to Duhamel's formula $$ x(t) = e^{(t - t_0) A} x_0 + \int_{t_0}^t e^{(t - s) A} b(s) , ds. $$

Counterexamples to common slips

The fundamental matrix $Φ (t)$ is a function of the single time variable $t$ , with $t_{0}$ fixed in the normalisation $Φ (t_{0}) = I_{n}$ . The propagator $Φ (t) Φ (s)^{- 1}$ is a function of two times $(t, s)$ . In the constant-coefficient case the propagator collapses to a function of $t - s$ alone; this is not the case for time-varying $A (t)$ , where $Φ (t) Φ (s)^{- 1}$ depends on $t$ and $s$ separately.
For time-varying $A (t)$ , the formula $Φ (t) = exp (\int_{t_{0}}^{t} A (s) d s)$ is false unless $A (t)$ commutes with $\int_{t_{0}}^{t} A (s) d s$ for every $t$ . This commutativity fails in general — for example, when $A (t) = diag (t, - t)$ shifted by a fixed off-diagonal contribution. The correct generalisation is the Magnus expansion or the time-ordered product, neither of which collapses to an ordinary matrix exponential of an integral.
The variation-of-constants identity is one-sided in time: it expresses $x (t)$ in terms of $x (t_{0})$ and the forcing on the interval $[t_{0}, t]$ (or $[t, t_{0}]$ when $t < t_{0}$ ). It does not require forcing data from outside this interval, and so applies equally to forward and backward propagation.

Key theorem with proof [Intermediate+]

Theorem (variation of constants; Lagrange 1809, Arnold ODE Ch.3 §19). Let $A : I \to Mat_{n} (R)$ and $b : I \to R^{n}$ be continuous on an open interval $I \subseteq R$ , $t_{0} \in I$ , $x_{0} \in R^{n}$ , and $Φ : I \to Mat_{n} (R)$ the fundamental matrix of the homogeneous system normalised at $t_{0}$ . The unique $C^{1}$ solution of the initial-value problem $$ \dot x(t) = A(t) x(t) + b(t), \qquad x(t_0) = x_0 $$ on $I$ is given by $$ x(t) = \Phi(t) \left( x_0 + \int_{t_0}^t \Phi(s)^{-1} b(s) , ds \right). $$

Proof. The argument has three steps: invertibility of $Φ$ , the ansatz, and uniqueness.

Step 1: $Φ (t)$ is invertible on $I$ . By Liouville's formula (proved below as a standalone proposition), $det Φ (t) = exp (\int_{t_{0}}^{t} tr A (s) d s)$ , which is positive for every $t \in I$ since the exponential of a real number is never zero. So $Φ (t)$ is invertible at every $t$ and $Φ^{- 1}$ is a $C^{1}$ matrix-valued function on $I$ with derivative $$ \frac{d}{dt} \Phi(t)^{-1} = -\Phi(t)^{-1} \dot\Phi(t) \Phi(t)^{-1} = -\Phi(t)^{-1} A(t) \Phi(t) \Phi(t)^{-1} = -\Phi(t)^{-1} A(t), $$ using the matrix-inverse derivative identity $\frac{d}{d t} (Φ^{- 1}) = - Φ^{- 1} \dot{Φ} Φ^{- 1}$ (which itself follows from differentiating $Φ Φ^{- 1} = I$ ).

Step 2: ansatz. Write the candidate solution as $x (t) = Φ (t) c (t)$ for some $C^{1}$ function $c : I \to R^{n}$ to be determined. Differentiate: $$ \dot x(t) = \dot\Phi(t) c(t) + \Phi(t) \dot c(t) = A(t) \Phi(t) c(t) + \Phi(t) \dot c(t) = A(t) x(t) + \Phi(t) \dot c(t). $$ The equation $\overset{x}{˙} = A (t) x + b (t)$ becomes $$ \Phi(t) \dot c(t) = b(t), \qquad \text{equivalently} \qquad \dot c(t) = \Phi(t)^{-1} b(t). $$ Integrate from $t_{0}$ to $t$ : $$ c(t) = c(t_0) + \int_{t_0}^t \Phi(s)^{-1} b(s) , ds. $$ The initial condition $x (t_{0}) = Φ (t_{0}) c (t_{0}) = I_{n} \cdot c (t_{0}) = c (t_{0}) = x_{0}$ fixes $c (t_{0}) = x_{0}$ . Substituting back, $$ x(t) = \Phi(t) \left( x_0 + \int_{t_0}^t \Phi(s)^{-1} b(s) , ds \right), $$ which is the claimed formula.

Step 3: uniqueness. Suppose $x$ and $\tilde{x}$ both solve the inhomogeneous initial-value problem. Their difference $y = x - \tilde{x}$ satisfies the homogeneous problem $\overset{y}{˙} = A (t) y$ , $y (t_{0}) = 0$ . By Picard-Lindelöf uniqueness applied to the homogeneous equation (the right-hand side is Lipschitz in $y$ with constant $sup_{[t_{0}, t]} ∥ A (s) ∥$ , which is finite on any bounded subinterval by continuity of $A$ ), the only solution is $y \equiv 0$ on $I$ . So $x = \tilde{x}$ . $□$

Proposition (Liouville's formula; Liouville 1838 ^{[Liouville 1838]}). Under the hypotheses above, $det Φ (t) = det Φ (t_{0}) exp (\int_{t_{0}}^{t} tr A (s) d s)$ . In particular, with $Φ (t_{0}) = I_{n}$ , $det Φ (t) = exp (\int_{t_{0}}^{t} tr A (s) d s)$ , which is positive and never vanishes.

Proof. Let $W (t) = det Φ (t)$ . Differentiate using Jacobi's formula for the derivative of a determinant: if $M (t)$ is a $C^{1}$ matrix-valued function, then $\frac{d}{d t} det M (t) = det M (t) \cdot tr (M (t)^{- 1} \dot{M} (t))$ wherever $M (t)$ is invertible. Apply this with $M = Φ$ and $\dot{Φ} = A Φ$ : $$ \dot W(t) = W(t) \cdot \mathrm{tr}(\Phi(t)^{-1} A(t) \Phi(t)) = W(t) \cdot \mathrm{tr}(A(t)), $$ using the trace identity $tr (B C B^{- 1}) = tr (C)$ which holds for any invertible $B$ and any $C$ of the same size. The scalar ODE $\dot{W} = tr (A (t)) \cdot W$ with $W (t_{0}) = det Φ (t_{0})$ integrates directly to give $W (t) = W (t_{0}) exp (\int_{t_{0}}^{t} tr A (s) d s)$ . $□$

Corollary (constant coefficients; Duhamel's formula). If $A (t) = A$ is constant, the fundamental matrix normalised at $t_{0}$ is $Φ (t) = e^{(t - t_{0}) A}$ , and the variation-of-constants solution becomes $$ x(t) = e^{(t - t_0) A} x_0 + \int_{t_0}^t e^{(t - s) A} b(s) , ds. $$

Proof. Direct: the matrix exponential $e^{(t - t_{0}) A}$ satisfies $\frac{d}{d t} e^{(t - t_{0}) A} = A e^{(t - t_{0}) A}$ by termwise differentiation of the power series (uniform convergence on bounded intervals justifies the term-by-term derivative), and $e^{0 \cdot A} = I_{n}$ . So $Φ (t) = e^{(t - t_{0}) A}$ is the normalised fundamental matrix. The propagator $Φ (t) Φ (s)^{- 1} = e^{(t - t_{0}) A} (e^{(s - t_{0}) A})^{- 1} = e^{(t - t_{0}) A} e^{- (s - t_{0}) A} = e^{(t - s) A}$ , using the commutativity of the exponential of $A$ with itself, gives the Duhamel form. $□$

Bridge. The variation-of-constants formula builds toward every linear analysis of forced systems in physics, engineering, and partial differential equations: every Duhamel-style formula in linear PDE theory is the infinite-dimensional generalisation of the same superposition principle, and every signal-processing convolution is its time-translation-invariant special case. The foundational reason the formula works is exactly the linearity of the equation: the difference between any two solutions of the inhomogeneous problem solves the homogeneous problem, so the solution set is an affine space modelled on the $n$ -dimensional vector space of homogeneous solutions. This is exactly the geometric content of "particular solution plus general homogeneous solution". The Wronskian determinant from [02.12.12] (first integrals) reappears here as Liouville's formula: $W (t) = det Φ (t)$ is a first integral of the homogeneous matrix equation $\dot{Φ} = A Φ$ when $tr A \equiv 0$ , and more generally its logarithmic time-derivative is $tr A (t)$ . The central insight is that the inhomogeneous problem is identified with the homogeneous propagator $Φ (t)$ acting on a time-varying initial condition $c (t)$ whose derivative $\overset{c}{˙} = Φ^{- 1} b$ records the cumulative effect of the forcing.

This pattern appears again in the constant-coefficient case as Duhamel's formula $x (t) = e^{(t - t_{0}) A} x_{0} + \int_{t_{0}}^{t} e^{(t - s) A} b (s) d s$ , which generalises to linear evolution equations on Banach spaces and to inhomogeneous linear PDEs via the corresponding $C^{0}$ -semigroup. Putting these together, the variation-of-constants framework is the bridge between the algebraic data $(A (t), b (t))$ and the analytic propagation of an arbitrary input through a linear system. The phase flow [02.12.02] underlying the homogeneous part composes with the integrated forcing to produce the full response, exactly identifying the inhomogeneous solution with a sum of the free flow $Φ (t) x_{0}$ and the convolution $\int Φ (t) Φ (s)^{- 1} b (s) d s$ of the propagator with the source.

Exercises [Intermediate+]

Exercise 4 (medium, symbolic).

Use Duhamel's formula to solve $\overset{x}{˙} = - x + cos t$ on $R$ with $x (0) = 0$ . Identify the transient and steady-state parts.

Hint

$x (t) = \int_{0}^{t} e^{- (t - s)} cos s d s$ . Integrate by parts twice or use complex exponentials.

Answer

By Duhamel, $x (t) = \int_{0}^{t} e^{- (t - s)} cos s d s = e^{- t} \int_{0}^{t} e^{s} cos s d s$ . Use $\int e^{s} cos s d s = \frac{1}{2} e^{s} (cos s + sin s)$ : $\int_{0}^{t} e^{s} cos s d s = \frac{1}{2} (e^{t} (cos t + sin t) - 1)$ . So $x (t) = \frac{1}{2} (cos t + sin t) - \frac{1}{2} e^{- t}$ . The transient is the $- \frac{1}{2} e^{- t}$ term (decays); the steady state is $\frac{1}{2} (cos t + sin t) = \frac{2}{2} cos (t - π /4)$ , a sinusoidal response of amplitude $\frac{2}{2}$ phase-shifted from the driving. Check: $\overset{x}{˙} = \frac{1}{2} (- sin t + cos t) + \frac{1}{2} e^{- t}$ ; $- x + cos t = - \frac{1}{2} cos t - \frac{1}{2} sin t + \frac{1}{2} e^{- t} + cos t = \frac{1}{2} cos t - \frac{1}{2} sin t + \frac{1}{2} e^{- t}$ , which equals $\overset{x}{˙}$ . Rubric: full credit for the integration, the transient/steady-state decomposition, and the phase-shifted amplitude $2 /2$ .

Exercise 5 (medium, short-answer).

Prove: the set of solutions of the homogeneous system $\overset{x}{˙} = A (t) x$ on $I$ is an $n$ -dimensional vector subspace of $C^{1} (I, R^{n})$ . Identify a basis.

Hint

Linearity gives a vector subspace; evaluation at $t_{0}$ gives a linear map to $R^{n}$ that is bijective by existence-uniqueness.

Answer

Let $V = {x \in C^{1} (I, R^{n}) : \overset{x}{˙} = A (t) x}$ . Linearity of the equation makes $V$ a vector subspace of $C^{1} (I, R^{n})$ . Consider the evaluation $ev_{t_{0}} : V \to R^{n}$ , $x \mapsto x (t_{0})$ . This is linear. By Picard-Lindelöf, the initial-value problem $\overset{x}{˙} = A (t) x$ , $x (t_{0}) = x_{0}$ has a unique solution for every $x_{0} \in R^{n}$ ; so $ev_{t_{0}}$ is bijective. Hence $V ≅ R^{n}$ as vector spaces, and $dim V = n$ . A basis: take ${Φ e_{1}, \dots, Φ e_{n}}$ , the columns of the fundamental matrix $Φ$ . These are linearly independent solutions (independent at $t_{0}$ by $Φ (t_{0}) = I_{n}$ , hence independent everywhere by injectivity of the linear isomorphism $ev_{t_{0}}^{- 1}$ ). Rubric: full credit for the dimension argument plus the columns-of- $Φ$ basis identification.

Exercise 6 (medium, symbolic).

For the scalar second-order equation $\overset{x}{¨} + ω_{0}^{2} x = f (t)$ , rewrite as a first-order $2 \times 2$ system, find the fundamental matrix for the homogeneous part, and write down the Duhamel integral.

Hint

State $y_{1} = x$ , $y_{2} = \overset{x}{˙}$ . The companion matrix is $A = (0 - ω_{0}^{2} 10)$ and the forcing is $b = (0, f (t))$ .

Answer

Set $y = (x, \overset{x}{˙})$ . The system is $\overset{y}{˙} = A y + b (t)$ with $A = (0 - ω_{0}^{2} 10)$ and $b (t) = (0, f (t))$ . The fundamental matrix is $Φ (t) = e^{t A} = (cos ω_{0} t - ω_{0} sin ω_{0} t ω_{0}^{- 1} sin ω_{0} t cos ω_{0} t)$ (verifiable by checking $Φ (0) = I_{2}$ and $\dot{Φ} = A Φ$ ). Duhamel gives $$ y(t) = e^{tA} y_0 + \int_0^t e^{(t-s)A} b(s) , ds = e^{tA} y_0 + \int_0^t \begin{pmatrix} \omega_0^{-1} \sin \omega_0 (t - s) \ \cos \omega_0 (t - s) \end{pmatrix} f(s) , ds. $$ The first component gives the classical Duhamel formula for the scalar second-order equation: $x (t) = (homogeneous) + ω_{0}^{- 1} \int_{0}^{t} sin (ω_{0} (t - s)) f (s) d s$ . Rubric: full credit for the companion matrix, the fundamental matrix, and the Duhamel integral.

Exercise 7 (hard, short-answer).

Prove Abel's identity in dimension $n = 2$ : for the second-order scalar equation $\overset{x}{¨} + p (t) \overset{x}{˙} + q (t) x = 0$ with linearly independent solutions $x_{1}, x_{2}$ , the Wronskian $W (t) = x_{1} \overset{x}{˙}_{2} - x_{2} \overset{x}{˙}_{1}$ satisfies $\dot{W} = - p (t) W$ , hence $W (t) = W (t_{0}) exp (- \int_{t_{0}}^{t} p (s) d s)$ .

Hint

Rewrite the scalar equation as a $2 \times 2$ first-order system and apply Liouville's formula. The trace of the companion matrix is $- p (t)$ .

Answer

Rewrite as $\overset{y}{˙} = A (t) y$ with $y = (x, \overset{x}{˙})$ and companion matrix $$ A(t) = \begin{pmatrix} 0 & 1 \ -q(t) & -p(t) \end{pmatrix}. $$ Trace: $tr A (t) = - p (t)$ . The fundamental matrix $Φ (t)$ has columns $(x_{1}, \overset{x}{˙}_{1})$ and $(x_{2}, \overset{x}{˙}_{2})$ , so $det Φ (t) = x_{1} \overset{x}{˙}_{2} - x_{2} \overset{x}{˙}_{1} = W (t)$ . Liouville's formula gives $det Φ (t) = det Φ (t_{0}) exp (\int_{t_{0}}^{t} tr A (s) d s) = W (t_{0}) exp (- \int_{t_{0}}^{t} p (s) d s)$ , hence $W (t) = W (t_{0}) exp (- \int_{t_{0}}^{t} p (s) d s)$ , equivalently $\dot{W} = - p (t) W$ . This is Abel's identity, the second-order scalar form of Liouville's determinant formula. Rubric: full credit for the companion matrix, the trace identification with $- p$ , and the application of Liouville.

Exercise 8 (hard, short-answer).

Show that the variation-of-constants formula extends to the inhomogeneous problem on a Banach space $E$ with $A (t)$ a continuous family of bounded linear operators on $E$ and $b : I \to E$ continuous. State the resulting formula and identify the appropriate notion of fundamental operator.

Hint

Replace matrices by bounded operators on $E$ . The fundamental operator $Φ (t) \in L (E)$ satisfies $\dot{Φ} = A (t) Φ$ , $Φ (t_{0}) = I_{E}$ . Existence-uniqueness follows from Picard-Lindelöf on the Banach space $L (E)$ .

Answer

Let $E$ be a Banach space, $A : I \to L (E)$ continuous with values in the bounded operators on $E$ , and $b : I \to E$ continuous. The fundamental operator $Φ : I \to L (E)$ is the unique $C^{1}$ operator-valued solution of $\dot{Φ} (t) = A (t) Φ (t)$ , $Φ (t_{0}) = I_{E}$ , which exists by Picard-Lindelöf applied to the operator equation in the Banach space $L (E)$ (the right-hand side is Lipschitz in $Φ$ with constant $sup_{[t_{0}, t]} ∥ A (s) ∥$ , finite on bounded intervals). Invertibility: $Φ (t)$ is invertible at every $t \in I$ because $Φ (t)$ satisfies the matrix-product semigroup-like property $Φ (t) = U (t, t_{0})$ where $U (t, s) U (s, t_{0}) = U (t, t_{0})$ (evolution-family axiom), so $U (t, t_{0})^{- 1} = U (t_{0}, t)$ exists.

The variation-of-constants formula is $$ x(t) = \Phi(t) \left( x_0 + \int_{t_0}^t \Phi(s)^{-1} b(s) , ds \right) = \Phi(t) x_0 + \int_{t_0}^t U(t, s) b(s) , ds, $$ where $U (t, s) = Φ (t) Φ (s)^{- 1}$ is the evolution family from time $s$ to time $t$ . The proof is the same ansatz $x = Φ c$ followed by integration of $\overset{c}{˙} = Φ^{- 1} b$ . Liouville's formula requires a notion of determinant; on finite-dimensional $E$ it carries over directly, on infinite-dimensional $E$ it is replaced by the Fredholm determinant when $A (t) - A_{0}$ is trace-class, or by the Carleman determinant in the Hilbert-Schmidt setting. The Banach-space framework is the foundation for $C^{0}$ -semigroup theory and inhomogeneous linear evolution equations on infinite-dimensional spaces, the natural home of inhomogeneous linear PDEs via Duhamel's principle. Rubric: full credit for the fundamental operator, the existence argument, the formula in evolution-family form, and the determinant-extension comment.

Lean formalization [Intermediate+]

Mathlib has the matrix exponential infrastructure (Mathlib.Analysis.NormedSpace.Exponential, Mathlib.LinearAlgebra.Matrix.MatrixExponential) and the Picard-Lindelöf existence-uniqueness theorem (Mathlib.Analysis.ODE.PicardLindelof), but does not package the fundamental matrix of a time-varying linear system or the variation-of-constants formula. The intended formalisation reads schematically:

import Mathlib.Analysis.ODE.PicardLindelof
import Mathlib.Analysis.NormedSpace.Exponential
import Mathlib.LinearAlgebra.Matrix.MatrixExponential

/-- The fundamental matrix of a homogeneous linear system $\dot x = A(t) x$
    normalised at $t_0$: the unique solution of $\dot\Phi = A(t)\Phi$,
    $\Phi(t_0) = I$. -/
noncomputable def fundamentalMatrix
    {n : Type*} [Fintype n] [DecidableEq n]
    (A : ℝ → Matrix n n ℝ) (hA : Continuous A) (t₀ : ℝ) :
    ℝ → Matrix n n ℝ :=
  sorry  -- apply ODE.IsPicardLindelof to the matrix equation column-by-column

/-- Liouville's formula for the determinant of the fundamental matrix. -/
theorem det_fundamentalMatrix
    {n : Type*} [Fintype n] [DecidableEq n]
    (A : ℝ → Matrix n n ℝ) (hA : Continuous A) (t₀ t : ℝ) :
    (fundamentalMatrix A hA t₀ t).det =
      Real.exp (∫ s in t₀..t, (A s).trace) :=
  sorry  -- Jacobi's formula plus the trace-conjugation identity

/-- Variation of constants for the inhomogeneous problem
    $\dot x = A(t) x + b(t)$, $x(t_0) = x_0$. -/
theorem variation_of_constants
    {n : Type*} [Fintype n] [DecidableEq n]
    (A : ℝ → Matrix n n ℝ) (hA : Continuous A)
    (b : ℝ → (n → ℝ)) (hb : Continuous b)
    (t₀ : ℝ) (x₀ : n → ℝ) :
    ∃ (x : ℝ → (n → ℝ)),
      (∀ t, HasDerivAt x (A t *ᵥ x t + b t) t) ∧
      x t₀ = x₀ ∧
      ∀ t, x t =
        (fundamentalMatrix A hA t₀ t) *ᵥ
        (x₀ + ∫ s in t₀..t, (fundamentalMatrix A hA t₀ s)⁻¹ *ᵥ b s) :=
  sorry  -- ansatz x = Φ c and integrate c' = Φ⁻¹ b

/-- Duhamel's formula in the constant-coefficient case. -/
theorem duhamel_constant_coefficient
    {n : Type*} [Fintype n] [DecidableEq n]
    (A : Matrix n n ℝ) (b : ℝ → (n → ℝ)) (hb : Continuous b)
    (t₀ : ℝ) (x₀ : n → ℝ) :
    ∃ (x : ℝ → (n → ℝ)),
      (∀ t, HasDerivAt x (A *ᵥ x t + b t) t) ∧
      x t₀ = x₀ ∧
      ∀ t, x t = Matrix.exp ℝ ((t - t₀) • A) *ᵥ x₀ +
                  ∫ s in t₀..t, Matrix.exp ℝ ((t - s) • A) *ᵥ b s :=
  sorry  -- specialise variation_of_constants to constant A

The proof gap is substantive but tractable. The fundamental-matrix construction is Picard-Lindelöf applied to the operator equation on $L (R^{n})$ , viewed as a Banach space under the operator norm; Mathlib already supplies the existence theorem. Liouville's formula requires Jacobi's formula for the derivative of a determinant (Matrix.det_deriv — partially in Mathlib via Matrix.det_mul-based identities but not yet packaged for time-varying matrices) plus the standard trace-conjugation identity. The variation-of-constants theorem is the ansatz $x = Φ c$ plus integration of $\overset{c}{˙} = Φ^{- 1} b$ , both within Mathlib's existing calculus on Banach-space-valued functions. Duhamel's formula is then a corollary in the constant-coefficient case via the matrix-exponential identity $e^{(t - t_{0}) A} \cdot e^{- (s - t_{0}) A} = e^{(t - s) A}$ , which Mathlib's Matrix.exp_add provides when $A$ commutes with itself.

Advanced results [Master]

Theorem (existence and uniqueness for the linear Cauchy problem; Cauchy 1820s, Picard 1893). Let $I \subseteq R$ be an open interval, $A : I \to Mat_{n} (R)$ and $b : I \to R^{n}$ continuous, $t_{0} \in I$ , and $x_{0} \in R^{n}$ . The inhomogeneous initial-value problem $\overset{x}{˙} = A (t) x + b (t)$ , $x (t_{0}) = x_{0}$ has a unique $C^{1}$ solution $x : I \to R^{n}$ defined on all of $I$ .

The proof is Picard iteration: define $x_{0} (t) \equiv x_{0}$ and $x_{k + 1} (t) = x_{0} + \int_{t_{0}}^{t} (A (s) x_{k} (s) + b (s)) d s$ . On any compact subinterval $[t_{0} - T, t_{0} + T] \subseteq I$ , the Lipschitz constant of the right-hand side in $x$ is $L = sup_{∣ t - t_{0} ∣ \leq T} ∥ A (t) ∥$ (finite by continuity), and the contraction-mapping theorem applied to the integral operator on $C ([t_{0} - T, t_{0} + T], R^{n})$ gives uniqueness and convergence of the iterates. The series $\sum_{k} ∥ x_{k + 1} - x_{k} ∥_{\infty}$ converges by the standard Picard-Lindelöf estimate $∥ x_{k + 1} - x_{k} ∥_{\infty} \leq (L T)^{k} / k!$ , so the limit exists and is continuous. Linearity ensures the solution extends to all of $I$ (no finite-time blow-up, in contrast to non-linear ODEs). The existence theorem in the linear case is global by virtue of the linear Lipschitz bound; Cauchy's 1820s polygon-method argument is the historical predecessor of the modern Picard iteration.

Theorem (Floquet's theorem; Floquet 1883). Let $A : R \to Mat_{n} (R)$ be continuous and periodic with period $T > 0$ : $A (t + T) = A (t)$ for every $t$ . The fundamental matrix $Φ (t)$ normalised at $t_{0} = 0$ admits the factorisation $$ \Phi(t) = P(t) e^{tR} $$ where $P : R \to Mat_{n} (C)$ is $T$ -periodic, $P (0) = I_{n}$ , and $R \in Mat_{n} (C)$ is a constant (possibly complex) matrix. The monodromy matrix $M = Φ (T) = e^{T R}$ encodes the period map; its eigenvalues are the Floquet multipliers and govern the stability of the system.

The proof uses the monodromy decomposition: define $M = Φ (T)$ (the period map) and choose any logarithm $R = T^{- 1} lo g M$ (existence requires complex logarithms; over $R$ one obtains $Φ (t) = P (t) e^{tR}$ with $P (t + 2 T) = P (t)$ in general). Then $P (t) = Φ (t) e^{- tR}$ satisfies $P (t + T) = Φ (t + T) e^{- (t + T) R} = Φ (t) M e^{- (t + T) R} = Φ (t) e^{T R} e^{- (t + T) R} = Φ (t) e^{- tR} = P (t)$ , where we used $Φ (t + T) = Φ (t) M$ from the periodicity of $A$ plus the uniqueness of the fundamental matrix. Floquet theory reduces the analysis of $T$ -periodic systems to the constant-coefficient analysis of the matrix $R$ , identifying the long-time behaviour with the spectrum of $M$ .

Theorem (variation of constants in evolution-family form; Hartman Ch.IV §5). Let $A : I \to Mat_{n} (R)$ be continuous and $U (t, s) = Φ (t) Φ (s)^{- 1}$ the propagator from time $s$ to time $t$ . Then $U$ satisfies the evolution-family axioms: $U (t, t) = I_{n}$ for every $t$ ; $U (t, s) U (s, r) = U (t, r)$ for every $r, s, t \in I$ ; $U$ is $C^{1}$ in both variables with $\partial_{t} U (t, s) = A (t) U (t, s)$ and $\partial_{s} U (t, s) = - U (t, s) A (s)$ . The unique solution of the inhomogeneous Cauchy problem $\overset{x}{˙} = A (t) x + b (t)$ , $x (t_{0}) = x_{0}$ is $$ x(t) = U(t, t_0) x_0 + \int_{t_0}^t U(t, s) b(s) , ds. $$

The evolution-family axioms generalise to time-varying linear systems the semigroup property $e^{(t - s) A} e^{(s - r) A} = e^{(t - r) A}$ of the matrix exponential. The propagator $U (t, s)$ is the time-varying analogue of $e^{(t - s) A}$ , and the integral $\int_{t_{0}}^{t} U (t, s) b (s) d s$ is the time-varying analogue of the Duhamel convolution.

Theorem (Duhamel principle for inhomogeneous linear PDEs). Let $A$ generate a $C^{0}$ -semigroup ${e^{t A}}_{t \geq 0}$ on a Banach space $E$ , and $b : [0, T] \to E$ a continuous source. The unique mild solution of the abstract Cauchy problem $\overset{u}{˙} = A u + b (t)$ , $u (0) = u_{0}$ is $$ u(t) = e^{t \mathcal{A}} u_0 + \int_0^t e^{(t - s) \mathcal{A}} b(s) , ds. $$

This is the infinite-dimensional Duhamel formula, the foundation of the modern theory of inhomogeneous linear evolution equations on Banach spaces. The hypotheses on $A$ (closedness, density of the domain, the Hille-Yosida resolvent estimate) are the abstract analogue of the Lipschitz continuity of $A (t)$ in the finite-dimensional setting. Applications include the heat equation, the wave equation, the Schrödinger equation, and the linear part of every semilinear evolution PDE.

Theorem (exponential dichotomy; Sacker-Sell 1976, Coppel 1978). A linear system $\overset{x}{˙} = A (t) x$ admits an exponential dichotomy on $I$ if there is a projection $P (t)$ commuting with the propagator and constants $K, α > 0$ such that $∥ U (t, s) P (s) ∥ \leq K e^{- α (t - s)}$ for $t \geq s$ and $∥ U (t, s) (I - P (s)) ∥ \leq K e^{- α (s - t)}$ for $t \leq s$ . Under exponential dichotomy, the inhomogeneous equation $\overset{x}{˙} = A (t) x + b (t)$ admits a unique bounded solution on $I$ for every bounded continuous $b$ , given by the dichotomy integral $$ x(t) = \int_{-\infty}^t U(t, s) P(s) b(s) , ds - \int_t^{\infty} U(t, s) (I - P(s)) b(s) , ds. $$

Exponential dichotomy is the time-varying generalisation of hyperbolicity for constant-coefficient systems (every eigenvalue has non-zero real part, with the projection $P$ onto the stable eigenspace). It is the foundational concept of non-autonomous dynamical systems and the home of the Sacker-Sell spectral theory, which generalises the spectrum of $A$ to the time-varying setting via the projector $P (t)$ and the exponential decay rates of the two pieces.

Theorem (Liouville's formula on a manifold; geometric form). Let $X$ be a smooth time-varying vector field on a smooth $n$ -manifold $M$ with flow $g^{t, s} : M \to M$ from time $s$ to time $t$ , and $Ω$ a smooth volume form on $M$ . Then $$ (g^{t, s})^* \Omega = \Omega \cdot \exp \left( \int_s^t \mathrm{div}\Omega X(g^{r, s} \cdot, r) , dr \right), $$ *where $\mathrm{div}\Omega X = \mathcal{L}_X \Omega / \Omega $. I n E u c l i d e an coor d ina t es w i t h$ \Omega = dx^1 \wedge \cdots \wedge dx^n $, t h e d i v er g e n cee q u a l s$ \sum_i \partial X^i / \partial x^i $, an df or a l in e a r sy s t e m$ X(x, t) = A(t) x $t h e d i v er g e n cee q u a l s$ \mathrm{tr}, A(t)$, recovering the determinant formula.*

The geometric Liouville formula identifies the determinant of the linear propagator with the volume distortion of the flow. The trace of $A (t)$ is the infinitesimal volume change rate; integrated, it gives the logarithm of the determinant of the propagator. In Hamiltonian mechanics, the divergence of a Hamiltonian vector field vanishes identically (preservation of the symplectic volume), recovering Liouville's classical theorem that the Hamiltonian flow preserves phase volume.

Theorem (impulse response and Green's function). For the constant-coefficient inhomogeneous problem $\overset{x}{˙} = A x + b (t)$ with $x (t_{0}) = 0$ , the response to an impulse $b (t) = δ (t - s) v$ (a Dirac measure at time $s$ with vector strength $v$ ) is $x (t) = e^{(t - s) A} v$ for $t > s$ and $x (t) = 0$ for $t < s$ . The Duhamel integral $\int_{t_{0}}^{t} e^{(t - s) A} b (s) d s$ is the convolution of the impulse-response kernel $G (t, s) = e^{(t - s) A} 1_{t \geq s}$ with the source $b$ , identifying the Duhamel formula as a Green's-function representation.

The impulse-response viewpoint connects the variation-of-constants formula to the theory of distributions and to the engineering literature on linear time-invariant systems. The Green's function $G (t, s)$ is the kernel of the inverse operator $(\partial_{t} - A)^{- 1}$ on the half-line, and the Duhamel formula is the explicit Green's-function representation of this inverse.

Synthesis. The variation-of-constants formula is the foundational reason every inhomogeneous linear ODE on a finite-dimensional space has its solution decomposed into a homogeneous propagation plus a convolution of the propagator with the source, and the central insight is that the inhomogeneous problem is identified with the homogeneous problem via the ansatz $x (t) = Φ (t) c (t)$ whose drift rate $\overset{c}{˙} = Φ (t)^{- 1} b (t)$ encodes the cumulative effect of the forcing. The constant-coefficient specialisation produces Duhamel's formula, in which the propagator $Φ (t) Φ (s)^{- 1} = e^{(t - s) A}$ depends only on the time difference and the inhomogeneous solution is a genuine convolution. Putting these together, the variation-of-constants framework is the bridge between the algebraic data $(A (t), b (t))$ and the analytic propagation of arbitrary inputs through a linear system, identifying the inhomogeneous Cauchy problem with the homogeneous propagator acting on a time-varying initial condition.

The theorem identifies several pictures that look distinct at first inspection. The variation-of-constants representation is identified with the evolution-family axiom $U (t, s) U (s, r) = U (t, r)$ ; this is exactly the algebraic content of "compose the propagators". The Wronskian determinant from [02.12.12] is identified with the determinant of the fundamental matrix, and Liouville's formula $det Φ (t) = exp (\int tr A)$ generalises the Wronskian identity to time-varying systems. The Duhamel convolution is identified with the Green's-function representation of the inverse operator $(\partial_{t} - A)^{- 1}$ , and the pattern recurs in linear PDE theory as the Duhamel principle for the inhomogeneous heat, wave, and Schrödinger equations. The bridge is the recognition that all of these pictures — variation of constants, evolution families, Duhamel, Green's functions, Liouville volume formula — are different presentations of the single fact that the inhomogeneous problem inverts the linear operator $\partial_{t} - A (t)$ via convolution with the homogeneous propagator. The exponential-dichotomy framework extends the picture to non-autonomous systems on the full line, and Floquet theory specialises it to periodic-coefficient systems via the monodromy matrix.

Full proof set [Master]

Theorem (variation of constants), proof. Stated and proved in the Intermediate-tier section: $Φ (t)$ is invertible by Liouville's formula; the ansatz $x = Φ c$ reduces the inhomogeneous equation to $\overset{c}{˙} = Φ^{- 1} b$ , integrating to $c (t) = x_{0} + \int_{t_{0}}^{t} Φ (s)^{- 1} b (s) d s$ ; uniqueness follows from Picard-Lindelöf on the difference, which solves the homogeneous problem with zero initial data. $□$

Proposition (Liouville's formula), proof. Stated and proved in the Intermediate-tier section: $\dot{W} = W \cdot tr (Φ^{- 1} A Φ) = W \cdot tr A$ by Jacobi's formula, integrating to the determinant identity. $□$

Proposition (Duhamel's formula in the constant-coefficient case), proof. Stated and proved in the Intermediate-tier section: $Φ (t) = e^{(t - t_{0}) A}$ is the normalised fundamental matrix by direct power-series differentiation, and the propagator $e^{(t - s) A}$ follows from the matrix-exponential semigroup identity. $□$

Proposition (existence and uniqueness for the linear Cauchy problem). Stated above. Proof. Define $x_{0} (t) \equiv x_{0}$ and $x_{k + 1} (t) = x_{0} + \int_{t_{0}}^{t} (A (s) x_{k} (s) + b (s)) d s$ . On a compact subinterval $J = [t_{0} - T, t_{0} + T] \subseteq I$ with $L = sup_{J} ∥ A (s) ∥$ and $M = sup_{J} ∥ b (s) ∥$ , induction gives $∥ x_{k + 1} (t) - x_{k} (t) ∥ \leq L \int_{t_{0}}^{t} ∥ x_{k} (s) - x_{k - 1} (s) ∥ d s$ , hence $∥ x_{k + 1} - x_{k} ∥_{\infty} \leq (L T)^{k + 1} / (k + 1)! \cdot (∥ x_{0} ∥ + M T)$ by induction starting from $∥ x_{1} - x_{0} ∥_{\infty} \leq T (L ∥ x_{0} ∥ + M)$ . The series $\sum_{k} (x_{k + 1} - x_{k})$ converges uniformly on $J$ by comparison with $\sum (L T)^{k} / k! = e^{L T}$ , so ${x_{k}}$ converges uniformly to a continuous limit $x$ , which is $C^{1}$ and satisfies the integral equation $x (t) = x_{0} + \int_{t_{0}}^{t} (A (s) x (s) + b (s)) d s$ . Differentiating gives $\overset{x}{˙} = A (t) x + b (t)$ with $x (t_{0}) = x_{0}$ . Uniqueness: two solutions $x, \tilde{x}$ have $y = x - \tilde{x}$ solving $\overset{y}{˙} = A (t) y$ , $y (t_{0}) = 0$ , and Grönwall's inequality applied to $∥ y (t) ∥^{2}$ gives $∥ y (t) ∥^{2} \leq ∥ y (t_{0}) ∥^{2} exp (2 \int_{t_{0}}^{t} ∥ A (s) ∥ d s) = 0$ , so $y \equiv 0$ . Linear growth of the iteration ensures convergence on the full interval $I$ with no blow-up; this is the special feature of the linear case relative to non-linear ODEs. $□$

Proposition (Floquet's theorem), proof sketch. Stated above. Proof sketch. Define $M = Φ (T)$ . The matrix $M$ is invertible (Liouville), and any invertible matrix has a logarithm in $Mat_{n} (C)$ ; set $R = T^{- 1} lo g M$ . Define $P (t) = Φ (t) e^{- tR}$ . Periodicity check: $Φ (t + T) = Φ (t) M = Φ (t) e^{T R}$ by uniqueness of the fundamental matrix (both sides satisfy $\dot{Ψ} = A (t + T) Ψ = A (t) Ψ$ with the same initial value at $t = 0$ shifted to $t = T$ ). Then $P (t + T) = Φ (t + T) e^{- (t + T) R} = Φ (t) e^{T R} e^{- (t + T) R} = Φ (t) e^{- tR} = P (t)$ , so $P$ is $T$ -periodic. The matrix $R$ is real if every Floquet multiplier (eigenvalue of $M$ ) is positive real; otherwise $R$ is complex and one obtains $Φ (t) = P (t) e^{tR}$ with $P (t + 2 T) = P (t)$ over $R$ (passing to the double cover). $□$

Proposition (evolution-family axioms), proof. Stated as part of Theorem 3 above. Proof. The composition $U (t, s) U (s, r) = Φ (t) Φ (s)^{- 1} Φ (s) Φ (r)^{- 1} = Φ (t) Φ (r)^{- 1} = U (t, r)$ by direct cancellation; $U (t, t) = Φ (t) Φ (t)^{- 1} = I_{n}$ ; smoothness in $(t, s)$ follows from the smoothness of $Φ$ and $Φ^{- 1}$ ; derivatives: $\partial_{t} U (t, s) = \dot{Φ} (t) Φ (s)^{- 1} = A (t) Φ (t) Φ (s)^{- 1} = A (t) U (t, s)$ and $\partial_{s} U (t, s) = Φ (t) \cdot \partial_{s} (Φ (s)^{- 1}) = Φ (t) \cdot (- Φ (s)^{- 1} A (s)) = - U (t, s) A (s)$ , using the matrix-inverse derivative identity. $□$

Proposition (Green's function for the constant-coefficient case). The Duhamel integral $\int_{t_{0}}^{t} e^{(t - s) A} b (s) d s$ is the convolution of the kernel $G (t - s) = e^{(t - s) A} 1_{t \geq s}$ with $b$ on $[t_{0}, \infty)$ .

Proof. The kernel $G (τ) = e^{τ A} 1_{τ \geq 0}$ has support on the half-line $τ \geq 0$ , and the convolution $(G * b) (t) = \int_{- \infty}^{\infty} G (t - s) b (s) d s = \int_{t_{0}}^{t} e^{(t - s) A} b (s) d s$ when $b$ is supported on $[t_{0}, \infty)$ . The convolution representation is the standard Green's-function format $x (t) = (G * b) (t)$ for the linear time-invariant operator $\partial_{t} - A$ with zero initial condition. The full inhomogeneous solution adds the homogeneous part $e^{(t - t_{0}) A} x_{0}$ to recover the variation-of-constants formula. $□$

Connections [Master]

Phase space, vector field, integral curve 02.12.01. The linear ODE $\overset{x}{˙} = A (t) x + b (t)$ is the special case of the general ODE $\overset{x}{˙} = X (x, t)$ in which the vector field is affine in the state variable: $X (x, t) = A (t) x + b (t)$ . The unit 02.12.01 provides the general framework — vector fields, integral curves, the Cauchy problem — within which the linear case admits the closed-form solution given by variation of constants. Linearity is the precise property that turns the abstract integral-curve construction into an algebraic formula.
Phase flow / one-parameter group 02.12.02. The fundamental matrix $Φ (t)$ is the matrix of the phase flow of the homogeneous linear vector field $X (x, t) = A (t) x$ evaluated against the standard basis. The propagator $U (t, s) = Φ (t) Φ (s)^{- 1}$ is the time-varying analogue of the one-parameter group $g^{t - s}$ of an autonomous flow, and the evolution-family axiom $U (t, s) U (s, r) = U (t, r)$ replaces the group law $g^{t - s} g^{s - r} = g^{t - r}$ . In the constant-coefficient case the two coincide: $U (t, s) = g^{t - s} = e^{(t - s) A}$ .
First integrals / conserved quantities 02.12.12. Liouville's formula identifies the determinant $W (t) = det Φ (t)$ as the exponential integral of the trace of $A (t)$ . When $tr A \equiv 0$ the determinant is constant: $W$ is a first integral of the homogeneous matrix equation $\dot{Φ} = A (t) Φ$ regarded as an ODE on $Mat_{n} (R)$ . The Wronskian-as-first-integral picture is the precise content shared between the present unit and 02.12.12. More generally, the energy-conservation analysis of Hamiltonian linear systems on $R^{2 n}$ uses the same trace-vanishing condition to identify the symplectic form as a preserved bilinear form.
Implicit and inverse function theorems 02.05.04. The existence and smoothness of the fundamental matrix follow from the Picard-Lindelöf existence theorem, whose proof uses the Banach fixed-point / contraction-mapping theorem on a function space. The implicit-function-theorem framework underlies the smooth-dependence-on-initial-conditions theorem for the linear ODE, and the differentiability of $Φ$ in its parameters (initial time, coefficient matrix entries) is the linear special case of the general parametric ODE smoothness theorem.
Rectification theorem 02.12.05. The constant-coefficient case $\overset{x}{˙} = A x$ with $A$ diagonalisable rectifies in eigenvector coordinates: in the basis of eigenvectors $v_{1}, \dots, v_{n}$ of $A$ , the system decouples into $n$ scalar equations $\overset{y}{˙}_{i} = λ_{i} y_{i}$ , each rectifiable as $\overset{y}{˙}_{i} = \partial / \partial y_{i}$ in logarithmic coordinates (for $λ_{i} \neq = 0$ ). The rectification theorem is the geometric local-normal-form statement of which the eigenvector-decomposition is the global linear-algebraic instance.
Lyapunov stability 02.12.08. The stability of an equilibrium of a non-linear system is analysed via the linearisation, which is exactly a constant-coefficient linear system $\overset{x}{˙} = A x$ with $A$ the Jacobian of the vector field at the equilibrium. The variation-of-constants formula gives the explicit response of the linearised system to perturbations, and the eigenvalue structure of $A$ governs the stability via Lyapunov's first method (hyperbolicity implies stability matching the linearisation, by Hartman-Grobman). The forced linear analysis in the present unit is the foundation of the perturbative stability theory of nonlinear equilibria.
Hamiltonian phase flow / symplectic geometry (forward). When $A$ is the Jacobian at the origin of a Hamiltonian vector field $X_{H}$ with $H$ quadratic, the system $\overset{x}{˙} = A x$ is itself Hamiltonian. The fundamental matrix $Φ (t)$ preserves the symplectic form $ω$ , and Liouville's formula in this setting gives $det Φ (t) = 1$ for every $t$ (the trace of a Hamiltonian matrix is the symplectic trace, which vanishes). The forced linear Hamiltonian system is the basis of perturbative celestial mechanics and the Lagrange-Laplace variation of constants for the orbital elements of a planet.

Historical & philosophical context [Master]

The variation-of-constants method appears in essentially its modern form in Lagrange's Mécanique Analytique (second edition, V $\overset{v}{^}$ e Courcier, Paris, 1811-1815) ^{[Lagrange 1811]}. Lagrange had applied the technique to the perturbed two-body problem in celestial mechanics from 1808 onward, treating the six Keplerian orbital elements as time-varying functions whose drift is controlled by the perturbing potential. The method generalised an earlier insight by Euler and Daniel Bernoulli that solutions of an inhomogeneous linear ODE can be sought as products of the homogeneous solution with a slowly varying coefficient. Lagrange's framing — "variation des constantes arbitraires" — became the standard 19th-century terminology, and the resulting formula for the inhomogeneous response is now the textbook starting point for forced linear systems.

The matrix-and-determinant viewpoint was assembled in stages through the early 19th century. Cauchy's École Polytechnique lectures of the 1820s, collected in his Œuvres complètes (Gauthier-Villars, Paris, Série II Tomes XI-XII) ^[Cauchy], gave the first rigorous existence proof for the linear initial-value problem via the polygon method (the historical predecessor of the modern Picard iteration), and identified the explicit linear-system Cauchy data $(A (t), b (t), x_{0}, t_{0})$ as the data determining the unique solution. Joseph Liouville's 1838 Note sur la théorie de la variation des constantes arbitraires (J. Math. Pures Appl. (1) 3, 342-349) ^{[Liouville 1838]} established the determinant formula $det Φ (t) = det Φ (t_{0}) exp (\int_{t_{0}}^{t} tr A (s) d s)$ , identifying the time evolution of the Wronskian determinant in terms of the trace of the coefficient matrix. The formula is the analytic backbone of the variation-of-constants method: it ensures the fundamental matrix is invertible at every time, which is what makes the ansatz $x = Φ c$ algebraically well-posed.

Jean-Marie Duhamel introduced the convolution formula for inhomogeneous linear problems in his 1833 Mémoire sur la méthode générale relative au mouvement de la chaleur dans les corps solides plongés dans des milieux dont la température varie avec le temps (J. École Polytechnique 14, 22e cahier, 20-77) ^{[Duhamel 1833]}, working on the inhomogeneous heat equation with time-varying boundary data. Duhamel's superposition principle — that the response to a continuous input is the integral of impulse responses against the source — became the foundational technique for inhomogeneous linear evolution equations, and the resulting Duhamel formula is the constant-coefficient (or, more generally, semigroup) specialisation of Lagrange's variation of constants. Émile Picard's 1893 Sur l'application des méthodes d'approximations successives à l'étude de certaines équations différentielles ordinaires (J. Math. Pures Appl. (4) 9, 217-271) ^{[Picard 1893]} gave the systematic iteration construction of the fundamental matrix, replacing Cauchy's polygon-method approach with the now-standard fixed-point construction on a function space. Picard's iteration, applied to the linear case, yields the matrix exponential as the limit of the polynomial partial sums $\sum_{k = 0}^{N} (t - t_{0})^{k} A^{k} / k!$ .

The 20th-century synthesis is in Hartman's Ordinary Differential Equations (Wiley 1964, second edition SIAM 2002) ^{[Hartman 2002]} and Coddington-Levinson's Theory of Ordinary Differential Equations (McGraw-Hill 1955) ^{[Coddington-Levinson 1955]}, both of which present the variation-of-constants framework in essentially its modern form, with Liouville's formula, Floquet theory for periodic coefficients, and exponential-dichotomy theory for non-autonomous systems. Arnold's Ordinary Differential Equations (Russian 1971, English 1973, third edition Springer 1992) ^[Arnold] integrates the variation-of-constants method into the geometric framework of phase flows on smooth manifolds, identifying the fundamental matrix with the linearisation of the flow and the Duhamel formula with the Green's-function representation of the linear evolution operator. The modern abstract framework — $C^{0}$ -semigroups on Banach spaces, evolution families for non-autonomous problems, the Hille-Yosida theorem — generalises the finite-dimensional theory to inhomogeneous linear PDEs and the abstract Cauchy problem, with the Duhamel formula as the universal solution representation ^{[Hartman 2002]}.

Bibliography [Master]

@book{Lagrange1811Mecanique,
  author    = {Lagrange, Joseph-Louis},
  title     = {M{\'e}canique Analytique},
  edition   = {2},
  publisher = {V$\hat{\text{v}}$e Courcier, Paris},
  year      = {1811--1815},
  note      = {Variation of arbitrary constants in the perturbed Keplerian problem}
}

@article{Liouville1838,
  author  = {Liouville, Joseph},
  title   = {Note sur la th{\'e}orie de la variation des constantes arbitraires},
  journal = {J. Math. Pures Appl. (1)},
  volume  = {3},
  year    = {1838},
  pages   = {342--349}
}

@book{CauchyOeuvres,
  author    = {Cauchy, Augustin-Louis},
  title     = {{\OE}uvres compl{\`e}tes},
  publisher = {Gauthier-Villars, Paris},
  series    = {S{\'e}rie II},
  volume    = {XI--XII},
  year      = {1882--1974},
  note      = {{\'E}cole Polytechnique lectures of the 1820s on the existence theorem for $\dot x = A(t) x + b(t)$ via the polygon method}
}

@article{Duhamel1833,
  author  = {Duhamel, Jean-Marie},
  title   = {M{\'e}moire sur la m{\'e}thode g{\'e}n{\'e}rale relative au mouvement de la chaleur dans les corps solides plong{\'e}s dans des milieux dont la temp{\'e}rature varie avec le temps},
  journal = {J. {\'E}cole Polytechnique},
  volume  = {14},
  number  = {22e cahier},
  year    = {1833},
  pages   = {20--77}
}

@article{Picard1893,
  author  = {Picard, {\'E}mile},
  title   = {Sur l'application des m{\'e}thodes d'approximations successives {\`a} l'{\'e}tude de certaines {\'e}quations diff{\'e}rentielles ordinaires},
  journal = {J. Math. Pures Appl. (4)},
  volume  = {9},
  year    = {1893},
  pages   = {217--271}
}

@article{Floquet1883,
  author  = {Floquet, Gaston},
  title   = {Sur les {\'e}quations diff{\'e}rentielles lin{\'e}aires {\`a} coefficients p{\'e}riodiques},
  journal = {Ann. {\'E}cole Norm. Sup. (2)},
  volume  = {12},
  year    = {1883},
  pages   = {47--88}
}

@book{Arnold1992ODE,
  author    = {Arnold, Vladimir I.},
  title     = {Ordinary Differential Equations},
  edition   = {3},
  publisher = {Springer-Verlag},
  year      = {1992},
  note      = {Translated from the Russian by Roger Cooke}
}

@book{Hartman2002,
  author    = {Hartman, Philip},
  title     = {Ordinary Differential Equations},
  edition   = {2},
  publisher = {SIAM},
  series    = {Classics in Applied Mathematics},
  volume    = {38},
  year      = {2002}
}

@book{CoddingtonLevinson1955,
  author    = {Coddington, Earl A. and Levinson, Norman},
  title     = {Theory of Ordinary Differential Equations},
  publisher = {McGraw-Hill},
  year      = {1955}
}

@book{Coppel1978,
  author    = {Coppel, W. A.},
  title     = {Dichotomies in Stability Theory},
  publisher = {Springer-Verlag},
  series    = {Lecture Notes in Mathematics},
  volume    = {629},
  year      = {1978}
}

@article{SackerSell1976,
  author  = {Sacker, Robert J. and Sell, George R.},
  title   = {A Spectral Theory for Linear Differential Systems},
  journal = {J. Differential Equations},
  volume  = {27},
  year    = {1978},
  pages   = {320--358}
}

@book{Strogatz2015NonlinearDynamics,
  author    = {Strogatz, Steven H.},
  title     = {Nonlinear Dynamics and Chaos},
  edition   = {2},
  publisher = {Westview Press},
  year      = {2015}
}

Prerequisites

02.05.04
02.12.01
02.12.02
02.12.12

Tier anchors

beginner: Arnold *Ordinary Differential Equations* Ch.3 §15 (linear systems as the simplest non-trivial ODE; matrix exponential as solution to the constant-coefficient homogeneous problem; the inhomogeneous picture as a forced version of the homogeneous flow); Strogatz *Nonlinear Dynamics and Chaos* §5.1 informal reading of forced linear oscillators and the resonance response
intermediate: Arnold *Ordinary Differential Equations* Ch.3 §§15-19 (fundamental matrix, Liouville's formula for the Wronskian determinant, variation of constants for the inhomogeneous system); Hartman *Ordinary Differential Equations* (2nd ed., SIAM Classics in Applied Mathematics 38, 2002) Ch.IV §§1-5; Coddington-Levinson *Theory of Ordinary Differential Equations* (McGraw-Hill 1955) Ch.3
master: Arnold *Ordinary Differential Equations* Ch.3 §§15-19 and Ch.4 §§24-29 (3rd ed., Springer 1992); Hartman *Ordinary Differential Equations* (2nd ed., SIAM 2002) Ch.IV (linear systems on Banach spaces, Floquet theory, exponential dichotomies); Coddington-Levinson *Theory of Ordinary Differential Equations* (McGraw-Hill 1955) Ch.3 (the classical variation-of-constants framework); Lagrange 1808-1809 *Mécanique Analytique* second edition (V$\hat{\text{v}}$e Courcier) — variation of arbitrary constants in the perturbed Keplerian problem; Cauchy 1820s École Polytechnique lectures (Œuvres complètes Série II Tomes XI-XII) — existence theorem in the linear case via the polygon method; Liouville 1838 *Note sur la théorie de la variation des constantes arbitraires* (J. Math. Pures Appl. (1) 3, 342-349) — the determinant formula $\det \Phi(t) = \det \Phi(t_0) \exp \int \mathrm{tr}\, A(s)\, ds$; Duhamel 1830s *Mémoire sur la méthode générale relative au mouvement de la chaleur dans les corps solides plongés dans des milieux dont la température varie avec le temps* (J. École Polytechnique 14, 22e cahier, 1833, pp. 20-77) — convolution formula for the inhomogeneous heat / wave problem; Picard 1893 *Sur l'application des méthodes d'approximations successives à l'étude de certaines équations différentielles ordinaires* (J. Math. Pures Appl. (4) 9, 217-271) — successive-approximation construction of the fundamental matrix and the general linear-existence theorem

References

TODO_REF
Arnold — Ordinary Differential Equations · Ch.3 §§15-19 fundamental matrix, Liouville's formula, variation of constants; Ch.4 §§24-29 inhomogeneous problem and Duhamel formula
TODO_REF
Hartman — Ordinary Differential Equations · Ch.IV §§1-5 linear systems on Banach spaces, fundamental matrix, Liouville's formula, variation-of-constants, Floquet theory
TODO_REF
Coddington-Levinson — Theory of Ordinary Differential Equations · Ch.3 classical variation-of-constants framework, $n$-th order scalar linear ODE via the companion matrix
TODO_REF
Lagrange — Mécanique Analytique · Seconde édition, V$\hat{\text{v}}$e Courcier, Paris, 1811-1815; the variation-of-arbitrary-constants method applied to perturbed celestial-mechanics equations
TODO_REF
Liouville — Note sur la théorie de la variation des constantes arbitraires · J. Math. Pures Appl. (1) 3 (1838) 342-349; the determinant formula $\det \Phi(t) = \det \Phi(t_0) \exp \int_{t_0}^t \mathrm{tr}\, A(s)\, ds$
TODO_REF
Cauchy — Œuvres complètes · Série II Tomes XI-XII (Gauthier-Villars, Paris); École Polytechnique lectures of the 1820s on the existence theorem for the linear ODE $\dot x = A(t) x + b(t)$ via the polygon method, predating the modern Picard iteration
TODO_REF
Duhamel — Mémoire sur la méthode générale relative au mouvement de la chaleur dans les corps solides · J. École Polytechnique 14 (22e cahier, 1833) 20-77; the convolution / superposition formula for inhomogeneous linear problems via integration of impulses against the homogeneous propagator
TODO_REF
Picard — Sur l'application des méthodes d'approximations successives · J. Math. Pures Appl. (4) 9 (1893) 217-271; successive-approximation construction of the fundamental matrix and the general existence theorem for linear ODEs
TODO_REF
Strogatz — Nonlinear Dynamics and Chaos · Ch.5 §§5.1-5.3 informal reading of forced linear oscillators, resonance response, and the steady-state amplitude formula for a driven damped harmonic oscillator

Reviewer

TBD

Estimated time

beginner: 15m
intermediate: 35m
master: 75m