Homogeneous Linear Equations with Constant Coefficients

Second-order linear differential equations with constant coefficients are among the most important and widely applicable ODEs. They appear in mechanics (oscillations), electrical engineering (circuits), and many other fields.

DefinitionSecond-Order Linear Homogeneous ODE

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

$ay'' + by' + cy = 0$

where $a, b, c$ are real constants with $a \neq 0$ . The equation is called homogeneous because the right-hand side is zero.

The solution method relies on seeking exponential solutions of the form $y = e^{rx}$ , which leads to the characteristic equation.

Characteristic Equation Method

Substituting $y = e^{rx}$ into the differential equation:

$ar^2e^{rx} + bre^{rx} + ce^{rx} = 0$

Factoring out $e^{rx}$ (which is never zero):

$ar^2 + br + c = 0$

This is the characteristic equation (or auxiliary equation). The nature of its roots determines the form of the general solution.

ExampleDistinct Real Roots

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

The characteristic equation is:

$r^2 - 5r + 6 = 0$

Factoring: $(r-2)(r-3) = 0$ , giving $r_1 = 2$ and $r_2 = 3$ .

The general solution is:

$y = c_1e^{2x} + c_2e^{3x}$

where $c_1$ and $c_2$ are arbitrary constants determined by initial conditions.

Three Cases Based on Discriminant

Let $\Delta = b^2 - 4ac$ be the discriminant. Three cases arise:

Case 1: Two Distinct Real Roots ( $\Delta > 0$ )

If $r_1 \neq r_2$ are real, the general solution is:

$y = c_1e^{r_1 x} + c_2e^{r_2 x}$

Case 2: Repeated Real Root ( $\Delta = 0$ )

If $r_1 = r_2 = r$ , the general solution is:

$y = (c_1 + c_2x)e^{rx}$

Case 3: Complex Conjugate Roots ( $\Delta < 0$ )

If $r = \alpha \pm i\beta$ , the general solution is:

$y = e^{\alpha x}(c_1\cos\beta x + c_2\sin\beta x)$

ExampleComplex Roots - Harmonic Oscillator

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

Characteristic equation: $r^2 + 4 = 0$

Roots: $r = \pm 2i$ (so $\alpha = 0$ , $\beta = 2$ )

General solution:

$y = c_1\cos 2x + c_2\sin 2x$

This describes undamped harmonic oscillation with angular frequency $\omega = 2$ .

Remark

The repeated root case requires the extra factor of $x$ to produce a second linearly independent solution. This can be verified using the method of reduction of order or Wronskian theory. Without this factor, we would have only one solution instead of the required two for a second-order equation.

ExampleRepeated Root - Critical Damping

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

Characteristic equation: $r^2 + 4r + 4 = 0$

Factoring: $(r+2)^2 = 0$ , giving repeated root $r = -2$ .

General solution:

$y = (c_1 + c_2x)e^{-2x}$

This represents critically damped motion in mechanical systems or circuits.

The characteristic equation method provides an algorithmic approach that always yields the complete solution for constant coefficient homogeneous equations.