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## 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 = ert where r is a constant. Differentiating and substituting into equation (1), we see that

a(ert)" + b(ert)' + cert = ar2ert + brert + cert = (ar2 + br + c)ert.

This shows that y(t) = ert 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) = ert 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.

Last Update: 2006-11-16