Differential Equations and Linear Algebra, 4th Edition

5.5 Nonhomogeneous Equations and Undetermined Coefficients

We learned in Section 5.3 how to solve homogeneous linear equations with constant coefficients, but we saw in Section 5.4 that an external force in a simple mechanical system contributes a nonhomogeneous term to its differential equation. The general nonhomogeneous nth-order linear equation with constant coefficients has the form

$$a_n{y}^{(n)}+a_{n-1}{y}^{(n-1)}+\cdots +a_{1}y\text{′}+a_{0}y=f(x).$$ (1)

By Theorem 5 of Section 5.2, a general solution of Eq. (1) has the form

$$y=y_c+y_p$$ (2)

where the complementary function