Second-Order Homogeneous Equations

You will learn how to solve second-order linear homogeneous differential equations with constant coefficients by using the characteristic equation, and understand the three possible cases for the roots.

A second-order linear homogeneous differential equation with constant coefficients has the form:

$$a y'' + b y' + c y = 0$$

We assume a solution of the form $y = e^{rx}$. Substituting this in gives the characteristic equation:

$$a r^2 + b r + c = 0$$

The form of the general solution depends on the roots of this quadratic:

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

The three panels show one solution curve from each case: exponential growth/decay (distinct real), modulated decay (repeated), and damped oscillation (complex).

Example 1: Distinct Real Roots

Solve $y'' - 5y' + 6y = 0$.

Solution: Characteristic equation: $r^2 - 5r + 6 = 0 \implies (r-2)(r-3) = 0$

Roots: $r = 2$, $r = 3$

General solution: $y = C_1 e^{2x} + C_2 e^{3x}$

Example 2: Repeated Root

Solve $y'' + 4y' + 4y = 0$.

Solution: Characteristic equation: $r^2 + 4r + 4 = (r+2)^2 = 0$

Repeated root $r = -2$

General solution: $y = (C_1 + C_2 x) e^{-2x}$

Example 3: Complex Roots

Solve $y'' + 2y' + 5y = 0$ with $y(0) = 1$, $y'(0) = 0$.

Solution: Characteristic equation: $r^2 + 2r + 5 = 0 \implies r = -1 \pm 2i$

General solution: $y = e^{-x} (C_1 \cos 2x + C_2 \sin 2x)$

Apply initial conditions to get $C_1 = 1$, $C_2 = \frac{1}{2}$.

Second-order homogeneous equations are the foundation for modeling oscillatory systems. They describe the motion of springs and masses (harmonic oscillators), the swing of a pendulum (for small angles), electrical circuits containing inductors and capacitors (LC circuits), and vibrations in mechanical structures. Understanding these helps engineers design stable bridges, car suspensions, and audio speakers.