Second Order Homogeneous Linear Differential Equation with Constant Coefficients

(1)

ay" + by' + cy = 0,

where a, b, and c are real constants and a, c ≠ 0.

Discussion If a = 0, the equation is a first order differential equation by' + cy = 0. If c = 0, the change of variables u = y' turns the given equation (1) into a first order differential equation au' + bu = 0. In each of these cases, the equation can be solved by the method of Section 14.1 or 14.2. (After finding u, y can be found by integration because y' = u.)

In Section 14.2 we found that the first order homogeneous linear differential equation with constant coefficients, y' + cy = 0, has the solution y = e^-ct. To get an idea of what to expect in the second order case, let us try to find a solution of equation (1) of the form y = e^rt where r is a constant. Differentiating and substituting into equation (1), we see that

a(e^rt)" + b(e^rt)' + ce^rt = ar²e^rt + bre^rt + ce^rt = (ar² + br + c)e^rt.

This shows that y(t) = e^rt is a solution of equation (1) if and only if

ar² + br + c = 0.

We should therefore expect that the solutions of the equation (1) will be built up from the functions y(t) = e^rt where r is a root of the polynomial az² + bz + c. We shall state the rule for finding the general solution of the equation (1) now and prove the rule later on.