First-order linear and separable ODEs

Intuition [Beginner]

A differential equation is a puzzle: you know something about the derivative of a function, and you must find the function itself. The simplest version says "the rate of change is proportional to the current value," which describes population growth, radioactive decay, and compound interest.

Some of these puzzles can be solved by moving all the $y$-terms to one side and all the $x$-terms to the other. This is called a separable equation. Other puzzles have a different form — they are linear in $y$ and its derivative — and they require a clever multiplying factor (called an integrating factor) to make them solvable.

Why does this concept exist? Because most physical laws are stated as relationships between a quantity and its rate of change, and solving these relationships is the first step to predicting how systems evolve over time.

Visual [Beginner]

The picture shows a slope field for the equation $y^{\prime }=y$. At each point $(x,y)$ in the plane, a short line segment is drawn with slope equal to $y$. The segments are flat along the horizontal axis ($y=0$), get steeper as $y$ increases, and point downward below the axis. Several solution curves thread through the field, each an exponential $y=C{e}^x$ for different starting values $C$.

Each solution curve is everywhere tangent to the slope field, confirming that its derivative at each point matches the prescribed slope.

Worked example [Beginner]

Solve the equation $y^{\prime }=3y$ given that $y=5$ when $x=0$.

Step 1. Separate the variables: move $y$ to the left and $x$ to the right. Write $dy/y=3dx$.

Step 2. Find the total of each side. The left side gives $\ln |y|$. The right side gives $3x+C$.

Step 3. Solve for $y$: $|y|=e^{3x+C}=e^C\cdot e^{3x}$, so $y=A{e}^{3x}$ where $A=\pm e^C$. Using the initial condition $y(0)=5$: $5=A\cdot e^0=A$, so $A=5$.

What this tells us: the solution $y=5e^{3x}$ grows by a factor of $e^3\approx 20$ for every unit increase in $x$. The initial value $5$ determines the scaling, and the rate $3$ determines the growth speed.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Definition (First-order ODE). A first-order ordinary differential equation is an equation of the form

$$\frac{dy}{dx}=f(x,y)$$

where $f:D\to \mathbb{R}$ is a function defined on a domain $D\subset {\mathbb{R}}^2$. A solution on an interval $I$ is a differentiable function $y:I\to \mathbb{R}$ satisfying $y^{\prime }(x)=f(x,y(x))$ for all $x\in I$.

Definition (Separable ODE). The equation $y^{\prime }=f(x,y)$ is separable if $f(x,y)=g(y)\cdot h(x)$ for some functions $g$ and $h$. The solution is obtained by writing $dy/g(y)=h(x)dx$ and antidifferentiating both sides.

Definition (Linear ODE). The equation $y^{\prime }+P(x)y=Q(x)$ is linear (first-order). When $Q(x)=0$ it is homogeneous; otherwise it is non-homogeneous. The integrating factor is $\mu (x)=e^{\int P(x)dx}$, and multiplying through by $\mu$ gives $(\mu y{)}^{\prime }=\mu Q$, which is solved by antidifferentiating.

Counterexamples to common slips

• Not every first-order ODE is separable. The equation $y^{\prime }=x+y$ cannot be separated into a product $g(y)\cdot h(x)$. It requires the integrating factor method for linear equations.

• Separation requires $g(y)\ne 0$. When solving $y^{\prime }=y(1-y)$ (the logistic equation), dividing by $y(1-y)$ excludes the constant solutions $y=0$ and $y=1$. These must be checked separately.

• The integrating factor is not $P(x)$ itself. The factor is $\mu (x)=e^{\int P(x)dx}$, not $P(x)$. The exponential of the integral is what makes the left side into a perfect derivative.

Key theorem with proof [Intermediate+]

Theorem (Picard-Lindelof existence and uniqueness). Let $f:D\to \mathbb{R}$ be continuous on the rectangle $R=\{(x,y):|x-x_0|\le a,|y-y_0|\le b\}$ and satisfy the Lipschitz condition in $y$: there exists $L>0$ such that

$$|f(x,y_1)-f(x,y_2)|\le L|y_1-y_2|$$

for all $(x,y_1),(x,y_2)\in R$. Then the initial value problem $y^{\prime }=f(x,y)$, $y(x_0)=y_0$ has a unique solution on the interval $[x_0-h,x_0+h]$ where $h=\min (a,b/M)$ and $M={\max }_R|f|$.

Proof (sketch via Picard iteration). Define the Picard iterates

$${\varphi }_0(x)=y_0,{\varphi }_{n+1}(x)=y_0+{\int }_{x_0}^{x}f(t,{\varphi }_n(t))dt.$$

Each ${\varphi }_n$ is well-defined on $[x_0-h,x_0+h]$ (by induction, using $|{\varphi }_n(x)-y_0|\le Mh\le b$). The key estimate is

$$|{\varphi }_{n+1}(x)-{\varphi }_n(x)|\le \frac{M{L}^n|x-x_0{|}^{n+1}}{(n+1)!}.$$

This is proved by induction: the base case $|{\varphi }_1(x)-{\varphi }_0(x)|\le M|x-x_0|$. For the inductive step, the Lipschitz condition gives

$$|{\varphi }_{n+2}(x)-{\varphi }_{n+1}(x)|\le {\int }_{x_0}^{x}L|{\varphi }_{n+1}(t)-{\varphi }_n(t)|dt\le L\cdot \frac{M{L}^n|t-x_0{|}^{n+1}}{(n+1)!}dt=\frac{M{L}^{n+1}|x-x_0{|}^{n+2}}{(n+2)!}.$$

Since $L^n{h}^n/n!\to 0$, the series $\sum |{\varphi }_{n+1}-{\varphi }_n|$ converges uniformly by comparison with $e^{Lh}$. Thus ${\varphi }_n$ converges uniformly to a limit $\varphi$ on $[x_0-h,x_0+h]$. Passing to the limit in the Picard recurrence gives $\varphi (x)=y_0+{\int }_{x_0}^{x}f(t,\varphi (t))dt$, so $\varphi$ solves the IVP.

For uniqueness: if $\psi$ is another solution, then $|\varphi (x)-\psi (x)|\le {\int }_{x_0}^{x}L|\varphi (t)-\psi (t)|dt$. By Gronwall's inequality (or directly by iteration), $\varphi =\psi$.

$\square$

Bridge. This theorem is the foundational reason that initial value problems for ODEs have well-defined solutions under mild conditions, and this is exactly the bridge from local analysis (continuity and Lipschitz bounds) to global existence. The central insight is that Picard iteration is a contraction in the supremum norm, so the Banach fixed-point theorem guarantees convergence. The result builds toward the general theory of second-order ODEs 02.08.02 where existence and uniqueness extend to systems, and appears again in 02.12.01 where the vector field formulation of $\dot{x}=f(x)$ produces integral curves via Picard iteration. Putting these together with the mean value theorem 02.05.02, the bridge is that Lipschitz continuity — the same condition underlying the MVT — is what makes the iteration contract.

Exercises [Intermediate+]

Advanced results [Master]

Theorem 1 (Exact equations). An ODE $M(x,y)dx+N(x,y)dy=0$ is exact if there exists $\psi (x,y)$ with ${\psi }_x=M$ and ${\psi }_y=N$. A necessary and sufficient condition (on a simply connected domain) is $M_y=N_x$. The solution is given implicitly by $\psi (x,y)=C$.

The proof of sufficiency constructs $\psi (x,y)={\int }_{x_0}^{x}M(t,y)dt+{\int }_{y_0}^{y}N(x_0,s)ds$ and verifies ${\psi }_x=M$ by the fundamental theorem and ${\psi }_y=N$ using the exactness condition $M_y=N_x$.

Theorem 2 (Integrating factors for non-exact equations). If $M_y\ne N_x$, an integrating factor $\mu (x,y)$ may exist such that $\mu Mdx+\mu Ndy=0$ is exact. If $(M_y-N_x)/N$ depends only on $x$, then $\mu (x)=e^{\int (M_y-N_x)/Ndx}$ works. If $(N_x-M_y)/M$ depends only on $y$, then $\mu (y)=e^{\int (N_x-M_y)/Mdy}$ works.

Theorem 3 (Bernoulli equation). The equation $y^{\prime }+P(x)y=Q(x)y^n$ (with $n\ne 0,1$) is transformed into a linear equation by the substitution $v=y^{1-n}$. The resulting linear equation $v^{\prime }+(1-n)P(x)v=(1-n)Q(x)$ is solved by the integrating factor method.

Theorem 4 (Clairaut equation). The equation $y=x{y}^{\prime }+f(y^{\prime })$ has general solution $y=Cx+f(C)$ (a family of straight lines) and singular solution obtained by eliminating $y^{\prime }$ from $y=x{y}^{\prime }+f(y^{\prime })$ and $x+f^{\prime }(y^{\prime })=0$. The singular solution is the envelope of the family of lines.

Theorem 5 (Gronwall's inequality). If $u(t)\le \alpha +{\int }_{t_0}^t\beta (s)u(s)ds$ with $\beta \ge 0$ continuous, then $u(t)\le \alpha e^{{\int }_{t_0}^t\beta (s)ds}$. This is the workhorse inequality for proving uniqueness of ODE solutions and for estimating the growth of perturbations.

Theorem 6 (Maximal interval of existence). If $f$ is locally Lipschitz in $y$ and continuous, the solution of $y^{\prime }=f(x,y)$, $y(x_0)=y_0$ extends uniquely to a maximal interval $(a,b)$. If $b<\infty$, then $|y(x)|\to \infty$ as $x\to b^{-}$ (or $y$ leaves the domain of $f$). This "blow-up criterion" characterises when solutions fail to exist globally.

Synthesis. The foundational reason first-order ODE theory works is that the Picard iteration is a contraction mapping in a suitable function space, and this is exactly the bridge from pointwise conditions (Lipschitz in $y$) to global existence (solutions on intervals). The central insight is that the three solution methods — separation of variables, integrating factors, and exact equations — are all special cases of finding a function $\psi (x,y)$ whose level curves are the solution trajectories. Putting these together with Gronwall's inequality, the bridge is that uniqueness (no two solutions cross) and existence (every initial condition generates a solution) together make the solution operator a well-defined flow map, which generalises to systems 02.08.02 and appears again in the phase-space formulation 02.12.01 where the flow map ${\varphi }_t$ moves points along integral curves. The pattern recurs in 02.12.13 where variation of parameters extends the integrating factor idea to inhomogeneous linear systems.

Full proof set [Master]

Proposition 1 (Solution of the linear ODE via integrating factor). The general solution of $y^{\prime }+P(x)y=Q(x)$ on an interval where $P$ and $Q$ are continuous is

$$y(x)=e^{-\int P(x)dx}\left(\int Q(x)e^{\int P(x)dx}dx+C\right).$$

Proof. Define $\mu (x)=e^{\int P(x)dx}$. Then

$$\mu y^{\prime }+\mu Py=(\mu y{)}^{\prime }$$

by the product rule and the fact that ${\mu }^{\prime }=P\mu$. Multiplying the ODE by $\mu$:

$$(\mu y{)}^{\prime }=\mu Q.$$

Antidifferentiating: $\mu y=\int \mu Qdx+C$, so $y={\mu }^{-1}\left(\int \mu Qdx+C\right)$. $\square$

Proposition 2 (Exactness criterion). On a simply connected domain $D$, the equation $Mdx+Ndy=0$ is exact if and only if $M_y=N_x$ on $D$.

Proof. ($\Rightarrow$) If exact, there exists $\psi$ with ${\psi }_x=M$ and ${\psi }_y=N$. By equality of mixed partials (valid since $M,N$ are $C^1$), $M_y={\psi }_{xy}={\psi }_{yx}=N_x$.

($\Leftarrow$) Define $\psi (x,y)={\int }_{x_0}^{x}M(t,y)dt+g(y)$ where $g$ is to be determined. Then ${\psi }_x=M(x,y)$ by FTC. For ${\psi }_y=N$: ${\psi }_y={\int }_{x_0}^x{M}_y(t,y)dt+g^{\prime }(y)$. Using $M_y=N_x$: ${\psi }_y={\int }_{x_0}^x{N}_x(t,y)dt+g^{\prime }(y)=N(x,y)-N(x_0,y)+g^{\prime }(y)$. Setting $g^{\prime }(y)=N(x_0,y)$ makes ${\psi }_y=N$. The function $g(y)={\int }_{y_0}^{y}N(x_0,s)ds$ works. $\square$

Proposition 3 (Bernoulli substitution). The substitution $v=y^{1-n}$ transforms $y^{\prime }+Py=Q{y}^n$ into the linear equation $v^{\prime }+(1-n)Pv=(1-n)Q$.

Proof. Since $v=y^{1-n}$, we have $v^{\prime }=(1-n)y^{-n}{y}^{\prime }$. Multiply the Bernoulli equation by $(1-n)y^{-n}$:

$$(1-n)y^{-n}{y}^{\prime }+(1-n)P{y}^{1-n}=(1-n)Q.$$

This is $v^{\prime }+(1-n)Pv=(1-n)Q$, a linear equation in $v$. $\square$

Connections [Master]

• Mean value theorem 02.05.02. The Lipschitz condition $|f(x,y_1)-f(x,y_2)|\le L|y_1-y_2|$ is a consequence of $f$ having a bounded partial derivative $f_y$, which in turn follows from the MVT applied to $f$ as a function of $y$. The MVT provides the quantitative estimate needed for the Picard iteration to converge.

• Fundamental theorems of calculus 02.04.04. The Picard iteration ${\varphi }_{n+1}(x)=y_0+{\int }_{x_0}^{x}f(t,{\varphi }_n(t))dt$ is built from the FTC: the solution is constructed as the integral of the right-hand side. The integrating factor method also relies on FTC when computing $\int P(x)dx$ and $\int \mu Qdx$.

• Phase space and integral curves 02.12.01. The first-order ODE $y^{\prime }=f(x,y)$ defines a vector field on the plane, and solutions are integral curves of this field. The Picard-Lindelof theorem guarantees that through each point passes exactly one such curve, establishing the flow map ${\varphi }_t$ that moves points along their trajectories.

Historical & philosophical context [Master]

Leibniz 1676, in his unpublished manuscripts on the calculus ^{[Leibniz1676]}, introduced the method of separation of variables as the first systematic technique for solving differential equations. His approach was purely algorithmic: isolate the variables and antidifferentiate. Bernoulli 1695, in Acta Eruditorum ^{[Bernoulli1695]}, posed the equation $y^{\prime }=P(x)y+Q(x)y^n$ as a challenge problem, now known as the Bernoulli equation. Leibniz himself solved it the following year by the substitution $v=y^{1-n}$.

Euler 1743, in De integratione aequationum differentialium ^[Euler1743], systematised the integrating factor method for linear equations and developed the theory of exact equations. Picard 1890, in the Memoires de la Societe des Sciences de Bordeaux ^[Picard1890], introduced the iteration scheme that bears his name and proved existence and uniqueness under Lipschitz conditions, using the method of successive approximations. Lindelof 1894 refined the existence theory by showing that the Lipschitz condition is the sharp hypothesis for uniqueness, distinguishing it from the weaker continuity hypothesis that guarantees existence alone (Peano's theorem, 1890).

Bibliography [Master]

@article{Bernoulli1695,
  author = {Bernoulli, Jakob},
  title = {Explicationes, annotationes et additiones ad ea quae in Actis superiorum annorum de Curva Elastica, Isochrona Paracentrica, et Caetera sunt prodita},
  journal = {Acta Eruditorum},
  year = {1695},
  pages = {537--553},
}

@article{Euler1743,
  author = {Euler, Leonhard},
  title = {De integratione aequationum differentialium},
  journal = {Novi Commentarii Academiae Scientiarum Petropolitanae},
  year = {1743},
}

@article{Picard1890,
  author = {Picard, Emile},
  title = {Memoire sur la theorie des equations aux derivees partielles et la methode des approximations successives},
  journal = {Journal de Mathematiques Pures et Appliquees},
  year = {1890},
  volume = {4},
  pages = {145--210},
}

@book{Apostol1967,
  author = {Apostol, Tom M.},
  title = {Calculus, Volume 1},
  publisher = {John Wiley \& Sons},
  year = {1967},
}

@book{BoyceDiPrima2008,
  author = {Boyce, William E. and DiPrima, Richard C.},
  title = {Elementary Differential Equations and Boundary Value Problems},
  publisher = {John Wiley \& Sons},
  year = {2008},
  edition = {9th},
}