I have to solve the ODE $$y''-5xy'+y=0$$ using power series.
I found
$$y(x)=a_{0}\left(1-\frac{1}{2!}x^{2}-\sum_{k=2}^{\infty}\frac{9\cdot19\cdot\ldots\cdot(10k-11)}{(2k)!}x^{2k}\right)+a_{1}\left(x+\sum_{k=1}^{\infty}\frac{4\cdot14\cdot\ldots\cdot(10k-6)}{(2k+1)!}x^{2k+1}\right)$$ and then I have to find the radius convergence.
Since there are two series, how should I do it?
With ratio test $$\lim_{n\to\infty}\left|\dfrac{a_{n+1}}{a_n}\right| = \lim_{n\to\infty}\left|\dfrac{\frac{9\cdot19\cdot\ldots\cdot(10k-11)(10k-1)}{(2k+2)!}x^{2k+2}}{\frac{9\cdot19\cdot\ldots\cdot(10k-11)}{(2k)!}x^{2k}}\right| = \lim_{n\to\infty}\left|\dfrac{10k-1}{(2k+1)(2k+2)}x^2\right|\to\left|\dfrac{10}{4}x^2\right|<1 $$ then the convergence interval is $(-\dfrac{2}{\sqrt{10}}<x<\dfrac{2}{\sqrt{10}})$.