Finding coefficients of a differential equation represented by power series I am studying for a discrete mathematics exam and have gotten stuck on this question:
Any function y of a real variable x that solves the differential equation:
$$\frac{d^4y}{dx^4} -16y =0$$
may be represented by a power series of the form: $$y = \sum_{n=0}^{+\infty} \frac{y_n}{n!} x^n$$
where the coefficients $y_0, y_1, y_2, y_3,...$ of this power series are real numbers. Find values of these coefficients $y_n$ for $n =0,1,2,3,4,...$     that yields a solution to the above differential equation with $y_0=1,$ $  y_1=0, $ $y_2=-4$ and $y_3=0$.
I have been researching this on the web but have been having trouble with this particular question. Can anyone help? I am still trying to learn this topic so would appreciate any insight into your thought process. Thank you very much. :)

Hint: $$y^{(4)}(x) = \sum_{n=0}^{+\infty} \frac{y_{n+4}}{n!}x^n.$$

 So $y_{n+4} = 16y_n$.

