Second-order linear ODEs with constant coefficients

Intuition [Beginner]

A second-order differential equation involves the second derivative — the rate of change of the rate of change. The physical prototype is a spring: a mass on a spring has a position $x$, a velocity $x^{\prime }$, and an acceleration $x^{\prime \prime }$. Newton's second law says the acceleration equals the force divided by the mass, and the spring force is proportional to displacement. This gives $x^{\prime \prime }=-kx$, a second-order ODE.

The equation $a{x}^{\prime \prime }+b{x}^{\prime }+cx=0$ with constant coefficients describes damped oscillations. The term $a{x}^{\prime \prime }$ is the inertia, $b{x}^{\prime }$ is the damping (friction), and $cx$ is the restoring force (spring). The key insight is that solutions are built from exponentials $e^{rx}$, where $r$ is determined by a quadratic equation (the characteristic equation).

Why does this concept exist? Because virtually every oscillatory system — springs, circuits, sound waves, structural vibrations — reduces to a constant-coefficient linear ODE, and the characteristic equation provides a clean, complete classification of all possible behaviours.

Visual [Beginner]

The picture shows three graphs stacked vertically, each a solution to $x^{\prime \prime }+b{x}^{\prime }+4x=0$ with different damping values $b$. The top graph ($b=0$) shows undamped oscillation: a pure sine wave. The middle graph ($b=2$) shows underdamped oscillation: a sine wave with exponentially shrinking amplitude. The bottom graph ($b=6$) shows overdamped decay: the solution returns to zero without oscillating.

The damping coefficient $b$ controls the transition from oscillation to decay. At the critical value $b=4$ (critical damping), the system returns to rest as fast as possible without oscillating.

Worked example [Beginner]

Solve $y^{\prime \prime }-5y^{\prime }+6y=0$.

Step 1. Write the characteristic equation: replace $y^{\prime \prime }$ with $r^2$, $y^{\prime }$ with $r$, and $y$ with $1$. This gives $r^2-5r+6=0$.

Step 2. Factor: $r^2-5r+6=(r-2)(r-3)=0$, so $r=2$ and $r=3$.

Step 3. The general solution is $y=C_1{e}^{2x}+C_2{e}^{3x}$, where $C_1$ and $C_2$ are arbitrary constants determined by initial conditions.

What this tells us: the solution is a sum of two exponentials, one growing twice as fast and one growing three times as fast. The specific mix is determined by where the system starts.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Definition (Second-order linear ODE with constant coefficients). A second-order linear homogeneous ODE with constant coefficients is

$$a{y}^{\prime \prime }+b{y}^{\prime }+cy=0$$

where $a,b,c\in \mathbb{R}$ and $a\ne 0$. The associated characteristic equation (or auxiliary equation) is the quadratic

$$a{r}^2+br+c=0.$$

The nature of the roots determines the form of the general solution:

• Distinct real roots $r_1\ne r_2$: $y=C_1{e}^{r_{1}x}+C_2{e}^{r_{2}x}$.
• Repeated real root $r$: $y=(C_1+C_{2}x)e^{rx}$.
• Complex conjugate roots $\alpha \pm \beta i$: $y=e^{\alpha x}(C_1\cos \beta x+C_2\sin \beta x)$.

The discriminant $\Delta =b^2-4ac$ determines which case applies: $\Delta >0$ (distinct real), $\Delta =0$ (repeated), $\Delta <0$ (complex).

For the non-homogeneous equation $a{y}^{\prime \prime }+b{y}^{\prime }+cy=g(x)$, the general solution is $y=y_h+y_p$ where $y_h$ is the general solution of the homogeneous equation and $y_p$ is any particular solution of the non-homogeneous equation.

Counterexamples to common slips

• Do not forget the $x$ factor in the repeated-root case. When $r$ is a repeated root, $e^{rx}$ alone gives only one independent solution. The second solution is $x{e}^{rx}$, not a second copy of $e^{rx}$.

• Complex roots produce real solutions. When $r=\alpha +\beta i$, the solution $e^{\alpha x}(C_1\cos \beta x+C_2\sin \beta x)$ is real-valued. The imaginary parts of $e^{(\alpha +\beta i)x}$ cancel when combined with its conjugate.

• The characteristic equation substitution is $y\to 1$, not $y\to r^0$. A common algebraic error is writing $a{r}^2+br+c=0$ when the constant term should be $c\cdot 1$, not $c\cdot r^0$. Both are correct, but the substitution rule must be applied consistently.

Key theorem with proof [Intermediate+]

Theorem (General solution via the characteristic equation). The general solution of $a{y}^{\prime \prime }+b{y}^{\prime }+cy=0$ on $\mathbb{R}$ is a two-dimensional vector space spanned by the solutions determined by the roots of $a{r}^2+br+c=0$. Specifically:

Case 1 (distinct real roots $r_1\ne r_2$). $y=C_1{e}^{r_{1}x}+C_2{e}^{r_{2}x}$.

Case 2 (repeated real root $r=-b/(2a)$). $y=(C_1+C_{2}x)e^{rx}$.

Case 3 (complex conjugate roots $\alpha \pm \beta i$). $y=e^{\alpha x}(C_1\cos \beta x+C_2\sin \beta x)$.

Proof of Case 1. The substitution $y=e^{rx}$ gives $y^{\prime }=r{e}^{rx}$, $y^{\prime \prime }=r^2{e}^{rx}$, so $a{y}^{\prime \prime }+b{y}^{\prime }+cy=(a{r}^2+br+c)e^{rx}=0$ if and only if $a{r}^2+br+c=0$ (since $e^{rx}\ne 0$). Thus $e^{r_{1}x}$ and $e^{r_{2}x}$ are both solutions. By linearity, $C_1{e}^{r_{1}x}+C_2{e}^{r_{2}x}$ is a solution for any $C_1,C_2$.

To see this is the general solution, we show the two solutions are linearly independent. The Wronskian is

$$W(e^{r_{1}x},e^{r_{2}x})=\left|\begin{array}{cc}{e}^{r_{1}x}& e^{r_{2}x}\\ r_1{e}^{r_{1}x}& r_2{e}^{r_{2}x}\end{array}\right|=(r_2-r_1)e^{(r_1+r_2)x}\ne 0$$

since $r_1\ne r_2$. By the Wronskian criterion, the solutions form a fundamental set.

$\square$

Proof of Case 2. One solution is $y_1=e^{rx}$. For a second linearly independent solution, use reduction of order: set $y_2=v(x)e^{rx}$. Then $y_2^{\prime }=(v^{\prime }+rv)e^{rx}$, $y_2^{\prime \prime }=(v^{\prime \prime }+2r{v}^{\prime }+r^{2}v)e^{rx}$. Substituting into the ODE:

$$a(v^{\prime \prime }+2r{v}^{\prime }+r^{2}v)+b(v^{\prime }+rv)+cv=0.$$

Since $a{r}^2+br+c=0$ and $2ar+b=0$ (both from $r=-b/(2a)$ being a double root), this simplifies to $a{v}^{\prime \prime }=0$, so $v^{\prime \prime }=0$, giving $v=C_1+C_{2}x$. Taking $v=x$ gives $y_2=x{e}^{rx}$, which is independent of $y_1=e^{rx}$.

$\square$

Proof of Case 3. With $r=\alpha \pm \beta i$, the complex solutions $e^{(\alpha +\beta i)x}$ and $e^{(\alpha -\beta i)x}$ span the solution space. By Euler's formula $e^{(\alpha +\beta i)x}=e^{\alpha x}(\cos \beta x+i\sin \beta x)$, the real and imaginary parts $e^{\alpha x}\cos \beta x$ and $e^{\alpha x}\sin \beta x$ are real-valued solutions. Their Wronskian is $W=\beta e^{2\alpha x}\ne 0$ (since $\beta \ne 0$ for complex roots), confirming linear independence.

$\square$

Bridge. This theorem is the foundational reason that constant-coefficient linear ODEs are completely solvable by algebra, and this is exactly the bridge from the quadratic formula to the full theory of linear differential equations. The central insight is that the characteristic polynomial encodes the entire solution structure: the discriminant determines whether solutions oscillate, decay monotonically, or grow. The result builds toward the Laplace transform 02.08.02 where the characteristic polynomial becomes the denominator of the transfer function, and appears again in the spring-mass system and electrical circuit applications where the three cases correspond to underdamped, critically damped, and overdamped behaviour. Putting these together with the Wronskian, the bridge is that linear independence of solutions — verified by a single determinant computation — guarantees completeness of the solution set.

Exercises [Intermediate+]

Advanced results [Master]

Theorem 1 (Superposition principle for non-homogeneous equations). If $y_{p_1}$ solves $a{y}^{\prime \prime }+b{y}^{\prime }+cy=g_1(x)$ and $y_{p_2}$ solves $a{y}^{\prime \prime }+b{y}^{\prime }+cy=g_2(x)$, then $y_{p_1}+y_{p_2}$ solves $a{y}^{\prime \prime }+b{y}^{\prime }+cy=g_1(x)+g_2(x)$. The general solution of the non-homogeneous equation is $y=y_h+y_p$ where $y_h$ spans the homogeneous solutions and $y_p$ is any particular solution.

Theorem 2 (Method of undetermined coefficients). For $g(x)=P_n(x)e^{\alpha x}\cos \beta x$ (or the sine version), guess $y_p=x^s{Q}_n(x)e^{\alpha x}(A\cos \beta x+B\sin \beta x)$ where $s$ is the multiplicity of $\alpha +\beta i$ as a root of the characteristic polynomial, $Q_n$ has the same degree as $P_n$, and $A,B$ are determined by substitution.

Theorem 3 (Variation of parameters). For $y^{\prime \prime }+p(x)y^{\prime }+q(x)y=g(x)$ with known fundamental solutions $y_1,y_2$, the particular solution is

$$y_p(x)=-y_1(x)\int \frac{y_2(x)g(x)}{W(x)}dx+y_2(x)\int \frac{y_1(x)g(x)}{W(x)}dx$$

where $W=y_1{y}_2^{\prime }-y_1^{\prime }{y}_2$ is the Wronskian. This method works for any continuous $g$, unlike undetermined coefficients which requires $g$ to be of a specific form.

Theorem 4 (Spring-mass-damper system). The equation $m{y}^{\prime \prime }+c{y}^{\prime }+ky=0$ with $m,c,k>0$ describes a damped harmonic oscillator. Define ${\omega }_0=\sqrt{k/m}$ (natural frequency) and $\zeta =c/(2\sqrt{mk})$ (damping ratio). Then: $\zeta <1$ (underdamped) gives oscillation with exponential decay; $\zeta =1$ (critically damped) gives fastest non-oscillatory return; $\zeta >1$ (overdamped) gives slow non-oscillatory return.

Theorem 5 (Laplace transform — preview). The Laplace transform $\mathcal{L}\{f\}(s)={\int }_0^{\infty }f(t)e^{-st}dt$ converts the ODE $a{y}^{\prime \prime }+b{y}^{\prime }+cy=g(t)$ into an algebraic equation in $s$ for $Y(s)=\mathcal{L}\{y\}$. The solution $y(t)$ is recovered by the inverse transform. The key property is $\mathcal{L}\{y^{\prime }\}(s)=sY(s)-y(0)$ and $\mathcal{L}\{y^{\prime \prime }\}(s)=s^{2}Y(s)-sy(0)-y^{\prime }(0)$, so initial conditions are built in automatically.

Theorem 6 (Electrical circuit analogy). An RLC circuit with resistance $R$, inductance $L$, and capacitance $C$ satisfies $L{q}^{\prime \prime }+R{q}^{\prime }+(1/C)q=V(t)$ where $q$ is the charge on the capacitor. This is identical in form to the spring-mass system with $m\leftrightarrow L$, $c\leftrightarrow R$, $k\leftrightarrow 1/C$. The characteristic equation $L{r}^2+Rr+1/C=0$ gives the same three cases for the transient response.

Synthesis. The foundational reason constant-coefficient linear ODEs are completely solvable is that the characteristic polynomial reduces the differential problem to an algebraic one, and this is exactly the bridge from calculus to algebra. The central insight is that $e^{rx}$ is an eigenfunction of differentiation ($d/dx(e^{rx})=r{e}^{rx}$), so the ODE becomes a polynomial in $r$. Putting these together with the Wronskian test for linear independence, the bridge is that two solutions span the full space whenever their Wronskian is nonzero — a single algebraic condition guarantees completeness. The pattern generalises to the Laplace transform where the algebraic variable $s$ replaces the derivative operator, and appears again in the first-order theory 02.08.01 where the integrating factor method is the $n=1$ case of the same characteristic-polynomial idea. The result builds toward the inhomogeneous theory 02.12.13 where variation of parameters extends the framework to arbitrary forcing functions, and the spring-mass and RLC applications show that the same algebra governs mechanical and electrical oscillations.

Full proof set [Master]

Proposition 1 (Reduction of order for repeated roots). If $r$ is a double root of $a{r}^2+br+c=0$, then $y_2=x{e}^{rx}$ is a solution of $a{y}^{\prime \prime }+b{y}^{\prime }+cy=0$ independent of $y_1=e^{rx}$.

Proof. Set $y=x{e}^{rx}$. Then $y^{\prime }=(1+rx)e^{rx}$, $y^{\prime \prime }=(2r+r^{2}x)e^{rx}$. Substituting:

$$a{y}^{\prime \prime }+b{y}^{\prime }+cy=[(2ar+br+c)+(a{r}^2+br+c)x]e^{rx}.$$

Since $r$ is a root: $a{r}^2+br+c=0$. Since $r=-b/(2a)$ is a double root: $2ar+b=0$, hence $2ar+br+c=2ar+b)r+(c-br+br)=(2ar+b)r+c$ — wait. More directly: $2ar+b=0$ because differentiating $a{r}^2+br+c$ gives $2ar+b=0$ at the double root. So $2ar+br+c=r(2ar+b)+c=0+c$. But also $c=a{r}^2+br=0$ at the double root... Let me redo this cleanly.

$$a{y}^{\prime \prime }+b{y}^{\prime }+cy=[a(2r+r^{2}x)+b(1+rx)+cx]e^{rx}=[(2ar+b)+(a{r}^2+br+c)x]e^{rx}.$$

The coefficient of $x$ is $a{r}^2+br+c=0$ (since $r$ is a root). The constant term is $2ar+b=0$ (since $r$ is a double root, i.e., $2ar+b=0$). Hence $a{y}^{\prime \prime }+b{y}^{\prime }+cy=0$.

The Wronskian $W(e^{rx},x{e}^{rx})=e^{rx}(e^{rx}+rx{e}^{rx})-r{e}^{rx}\cdot x{e}^{rx}=e^{2rx}\ne 0$. $\square$

Proposition 2 (Abel's identity). For $y^{\prime \prime }+p(x)y^{\prime }+q(x)y=0$, the Wronskian $W(x)=y_1{y}_2^{\prime }-y_1^{\prime }{y}_2$ of any two solutions satisfies $W^{\prime }(x)=-p(x)W(x)$.

Proof.

$$W^{\prime }=y_1{y}_2^{\prime \prime }+y_1^{\prime }{y}_2^{\prime }-y_1^{\prime \prime }{y}_2-y_1^{\prime }{y}_2^{\prime }=y_1{y}_2^{\prime \prime }-y_1^{\prime \prime }{y}_2.$$

From the ODE: $y_1^{\prime \prime }=-p{y}_1^{\prime }-q{y}_1$ and $y_2^{\prime \prime }=-p{y}_2^{\prime }-q{y}_2$. Substituting:

$$W^{\prime }=y_1(-p{y}_2^{\prime }-q{y}_2)-(-p{y}_1^{\prime }-q{y}_1)y_2=-p(y_1{y}_2^{\prime }-y_1^{\prime }{y}_2)=-pW.$$

The solution of $W^{\prime }=-pW$ is $W(x)=W(x_0)e^{-{\int }_{x_0}^{x}p(t)dt}$. $\square$

Proposition 3 (Variation of parameters formula). For $y^{\prime \prime }+p(x)y^{\prime }+q(x)y=g(x)$ with independent solutions $y_1,y_2$ of the homogeneous equation, $y_p=u_1{y}_1+u_2{y}_2$ is a particular solution where $u_1^{\prime }=-y_{2}g/W$ and $u_2^{\prime }=y_{1}g/W$.

Proof. Set $y_p=u_1{y}_1+u_2{y}_2$ and require $u_1^{\prime }{y}_1+u_2^{\prime }{y}_2=0$ (one condition to simplify). Then $y_p^{\prime }=u_1{y}_1^{\prime }+u_2{y}_2^{\prime }$ and $y_p^{\prime \prime }=u_1{y}_1^{\prime \prime }+u_2{y}_2^{\prime \prime }+u_1^{\prime }{y}_1^{\prime }+u_2^{\prime }{y}_2^{\prime }$. Substituting into the ODE:

$$u_1(y_1^{\prime \prime }+p{y}_1^{\prime }+q{y}_1)+u_2(y_2^{\prime \prime }+p{y}_2^{\prime }+q{y}_2)+u_1^{\prime }{y}_1^{\prime }+u_2^{\prime }{y}_2^{\prime }=g.$$

The first two groups vanish (homogeneous equation). So $u_1^{\prime }{y}_1^{\prime }+u_2^{\prime }{y}_2^{\prime }=g$. Combined with $u_1^{\prime }{y}_1+u_2^{\prime }{y}_2=0$, this is a $2\times 2$ linear system for $(u_1^{\prime },u_2^{\prime })$ with determinant $W\ne 0$. Solving: $u_1^{\prime }=-y_{2}g/W$, $u_2^{\prime }=y_{1}g/W$. $\square$

Connections [Master]

• First-order linear and separable ODEs 02.08.01. The second-order theory generalises the first-order integrating factor method: the characteristic equation $a{r}^2+br+c=0$ is the "polynomial upgrade" of the first-order integrating factor. Reduction of order (finding a second solution from a known one) parallels the first-order technique of finding an integrating factor from a known solution.

• Mean value theorem 02.05.02. The existence and uniqueness theory for second-order ODEs relies on the Picard-Lindelof theorem applied to the equivalent first-order system $(y,y^{\prime })$, which in turn depends on the Lipschitz conditions established via the MVT. The Wronskian's exponential decay (Abel's identity) is itself a first-order ODE solved by an integrating factor.

• Complex numbers and Euler's formula 02.09.01. The complex-root case of the characteristic equation produces solutions $e^{\alpha x}\cos \beta x$ and $e^{\alpha x}\sin \beta x$ via Euler's formula $e^{(\alpha +\beta i)x}=e^{\alpha x}(\cos \beta x+i\sin \beta x)$. Without the algebra of complex numbers, the oscillatory case cannot be expressed in closed form. The hyperbolic functions 02.06.04 handle the analogous real-root case where the exponentials are purely real.

Historical & philosophical context [Master]

Euler 1743, in De integratione aequationum differentialium ^[Euler1743], introduced the method of assuming $y=e^{rx}$ and reducing the ODE to a polynomial equation in $r$. This was the first systematic technique for constant-coefficient linear equations and established the characteristic equation as the central object. Euler immediately recognised the three cases (distinct real, repeated, complex) and showed that complex roots produce trigonometric solutions.

d'Alembert 1747, in Recherches sur la vibration des cordes ^{[DAlembert1747]}, developed the reduction-of-order technique while studying the wave equation, and was the first to articulate the principle that the general solution of a linear equation is the sum of the homogeneous and particular solutions. Lagrange 1775, in Nouvelles recherches sur la nature et la propagation du son ^{[Lagrange1775]}, introduced variation of parameters as a systematic method for finding particular solutions of non-homogeneous equations, extending d'Alembert's superposition principle to arbitrary forcing functions.

Laplace 1782, in his Memoires de l'Academie royale des Sciences de Paris ^{[Laplace1782]}, introduced the integral transform that converts differential equations into algebraic equations. The Laplace transform builds directly on the characteristic-equation idea — the transform variable $s$ plays the same algebraic role as the characteristic root $r$ — while also incorporating initial conditions automatically.

Bibliography [Master]

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

@article{DAlembert1747,
  author = {d'Alembert, Jean le Rond},
  title = {Recherches sur la courbe que forme une corde tendue mise en vibration},
  journal = {Histoire de l'Acad\'emie Royale des Sciences et Belles-Lettres de Berlin},
  year = {1747},
  pages = {214--249},
}

@article{Lagrange1775,
  author = {Lagrange, Joseph-Louis},
  title = {Nouvelles recherches sur la nature et la propagation du son},
  journal = {Miscellanea Taurinensia},
  year = {1775},
}

@article{Laplace1782,
  author = {Laplace, Pierre-Simon},
  title = {Th\'eorie des attractions des sph\'eroides et de la figure des plan\`etes},
  journal = {M\'emoires de l'Acad\'emie Royale des Sciences de Paris},
  year = {1782},
}

@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},
}