What is the general solution of the differential equation of the form $y^{(4)} + ay = f(x)$, where $f(x)$ is a polynomial of $x$.
In my textbook, I have found the method of finding the general solution of high order differential equations of the form y^{(n)} = g(y^{(n-1)}, \cdots, y', x) or Euler equations. But this equation is of neither kind.
It is not difficult to find a particular solution. For example, when $a \neq 0$ and $f(x)$ is of degree $2$, $f(x) = b_2x^2 + b_1 x + b_0$. Then $y = \frac{1}{a}(b_2x^2 + b_1 x + b_0 + ae ^{\sqrt[4]{-a}x})$ is a particular solution.
But what will the general solution be like?
I am doing algebra everyday, so I know little about differential equations. Thanks to everyone for viewing or help.