Inhomogeneous differential equation

Question

How would you solve this:

Let $p(x)=\alpha x+\beta$ be a first degree polynomial where $\alpha$ and $\beta$ are arbitrary real numbers. Show that there is a first degree polynomial $q$ that solves the inhomogeneous differential equation $a_ny^{(n)}+a_{n-1}y^{(n-1)}+a_{n-2}y^{(n-2)}+...+a_1y'+a_0y=p$, where $a_0\neq 0$.

Thanks on beforehand

Answer 1

Just plug in $q(x)=ax+b$ in the equation and determine $a$ and $b$ by comparing coefficients.. The answer is $q(x)=\frac {\alpha} {a_0} x+\frac {\beta a_0 -\alpha a_1} {a_0^{2}}$.