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

P0(x)y(n)+P1(x)y(n1)++Pn1(x)y+Pn(x)y=F(x). (1)

Unless otherwise noted, we will always assume that the coefficient functions Pi(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 P0(x)0 at each point of I, we can divide each term in Eq. (1) by P0(x) to obtain an equation with leading coefficient 1, of the form

y(n)+p1(x)y(n1)+

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