We have the ODE:
$$y' = -3y$$
Assume that the solution of the ODE can be written as a Taylor series in the form:
$$y(x) = a_0 + a_1x + a_2x^2 + \ldots + a_nx^n + \ldots$$
Using this infinite series, find an equation that relates $a_{n+1}$ to $a_n$
From that result, deduce the coefficients $a_1, a_2, a_3, \ldots, a_n$ in terms of $a_0$.
I have tried substituting $y(x)$ as well as the derivative of $y(x)$ into the ODE but am unsure what to do next to relate $a_{n+1}$ to $a_n$.
Any help would be greatly appreciated.
EDIT
I think the answer to the first part is (correct me if I'm wrong) :
$$a_{n+1} = -\frac{3a_n}{n+1}$$
I then used this to find the first few terms ($a_1,a_2,a_3$ etc.) but can't seem to figure out the second part of the problem (deduce the coefficients $a_n$ in terms of $a_0$)