Hello everyone!

Here's a DE: $\displaystyle y' -5x^4y=3x^9$.

Letting $\displaystyle y=\sum_{n=0}^{\infty} c_n x^n$ and after some algebra, we get:

$\displaystyle c_1+2c_2x+3c_3x^2+4c_4x^3-3x^9$ $\displaystyle +\sum_{k=0}^{\infty}(\left (k+5)c_{k+5}-5c_k \right)\,x^{k+4}=0$.

does this mean that $\displaystyle c_1=c_2=c_3=c_4=0$?

what about $\displaystyle -3x^9$? And is the relation $\displaystyle (k+5)c_{k+5}-5c_k$ even valid.

Solved the DE w\out power series, and I checked it, I got $\displaystyle y=-\frac{3}{5}(x^5+1)$

How to do the algebra in the series solution?

Thanks!