# 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

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

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