Differential Equations and Linear Algebra, 4th Edition

5.2 General Solutions of Linear Equations

We now show that our discussion in Section 5.1 of second-order linear equations generalizes in a very natural way to the general nth-order linear differential equation of the form

$$P_0(x)y^{(n)}+P_1(x)y^{(n-1)}+\cdots +P_{n-1}(x)y\text{′}+P_n(x)y=F(x).$$ (1)

Unless otherwise noted, we will always assume that the coefficient functions $P_i(x)$ and F(x)are continuous on some open interval I (perhaps unbounded) where we wish to solve the equation. Under the additional assumption that $P_0(x)\ne 0$ at each point of I, we can divide each term in Eq. (1) by $P_0(x)$ to obtain an equation with leading coefficient 1, of the form

$$y^{(n)}+p_1(x)y^{(n-1)}+$$