Homogeneous Second Order Differential Equations with Constant Coefficients

A differential equation is homogeneous if the constant term is 0. A linear, homogeneous second order differential equation with constant coefficients can be written in the form

y^{″} + b y^{'} + c y = 0

By guessing the solution $y = e^{r x}$ , you get the characteristic equation

r^{2} + b r + c = 0

Two Real Solutions

When the characteristic equation has two solutions ( $r_{1}$ and $r_{2}$ ), the general solution is:

y = C_{1} e^{r_{1} x} + C_{2} e^{r_{2} x}

Example 1

Solve the differential equation $y^{″} + y^{'} - y = 0$

The characteristic equation is

\begin{array}{l} r^{2} + r - 1 & = 0 \\ (r + 2) (r - 1) & = 0 \end{array}

That gives you the solutions $r_{1} = 1$ and $r_{2} = - 2$ . Enter them into the general formula and get

y (x) = C_{1} e^{x} + C_{2} e^{- 2 x}

When the characteristic equation has one solution $r_{1} = r_{2}$ , the general solution can be written as

y = C_{1} e^{r x} + C_{2} x e^{r x}

Example 2

Solve the differential equation $y^{″} + 2 y^{'} + y = 0$

The characteristic equation is

\begin{array}{l} r^{2} + 2 r + 1 & = 0 \\ (r + 1) (r + 1) & = 0 \end{array}

This gives you the solution $r = r_{1} = r_{2} = 1$ . Enter $r$ into the formula for the general solution and get

y (x) = C_{1} e^{x} + C_{2} x e^{x}

When $r_{1}$ and $r_{2}$ are complex numbers, the general solution can be written as

y = e^{A x} (C_{1} \sin B x + C_{2} \cos B x)

where $r_{1} = A + B i$ and $r_{2} = A - B i$ . Recall that $i = \sqrt{- 1}$ .

Example 3

Solve the differential equation $y^{″} - 2 y^{'} + y = 0$

Use the characteristic equation. It is

r^{2} - 2 r + 5 = 0

That gives you

\begin{array}{l} r & = \frac{2 \pm \sqrt{4 - 4 \cdot 1 \cdot 5}}{2} \\ = \frac{2 \pm \sqrt{- 16}}{2} \\ = \frac{2 \pm 4 i}{2} \\ = 1 \pm 2 i \end{array}

Now you get $A = 1$ and $B = 2$ . Enter these into the formula for the general solution and you’ll get

y (x) = e^{x} (C_{1} \sin (2 x) + C_{2} \cos (2 x))

It’s useful to know these formulas by heart!