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Home Differential Equations Second Order Homogeneous Linear Equations Second Order Homogeneous Linear Differential Equation with Constant Coefficients  
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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 = e^{rt} where r is a constant. Differentiating and substituting into equation (1), we see that a(e^{rt})" + b(e^{rt})' + ce^{rt} = ar^{2}e^{rt} + bre^{rt} + ce^{rt} = (ar^{2} + br + c)e^{rt}. This shows that y(t) = e^{rt} is a solution of equation (1) if and only if ar^{2} + 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 az^{2} + bz + c. We shall state the rule for finding the general solution of the equation (1) now and prove the rule later on.


Home Differential Equations Second Order Homogeneous Linear Equations Second Order Homogeneous Linear Differential Equation with Constant Coefficients 