Second Order Linear Differential Equation with Constant Coefficients

(1)

ay" + by +cy=f(t)

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

A differential equation of this form describes a mass-spring system where an outside force f(t) is applied to the mass. The function f(t) is called the forcing term.

As before, if a = 0 the equation is a first order linear differential equation in y, and if c = 0 it is a first order linear differential equation in y'. fn each of these cases, the equation should be solved by the methods of Section 14.3. Hereafter we assume a, c ≠ 0.

To get started, let us review the first theorem on first order linear differential equations. Theorem 1 in Section 14.3 states that the general solution of a first order linear differential equation is the sum

y(t) + Bx(t)

where y(t) is a particular solution of the given equation and x(t) is a particular solution of the corresponding homogeneous equation. Here is a similar theorem for second order equations.

As in the first order case, this theorem is proved using the Principle of Superposition.