Series Solution to a 2nd order ODE I am very stuck on a homework problem involving series solutions and 2nd order ODE. Could anyone point me towards a solution?
Consider the ODE 
$$
xy''  +  y'  -  y  =  0
$$
0 is a singular point for the differential equation, but there is a solution that is analytic at 0. Find the series representation centered at 0 for this solution. 
Any help at all would be greatly appreciated.
 A: $$xy'' +y' -y=0\qquad ......(1)$$
$~x=0~$ is a regular singular point of equation $(1)$.
So the equation admits of a Frobenius series of the form $$y=\sum_{n=0}^{\infty}C_n~x^{n+r},\qquad C_0\neq 0 \qquad ..........(2)$$ 
which converges for all $~x~$.
From $(2)$,
$$y'(x)=\sum_{n=0}^{\infty}(n+r)C_n~x^{n+r-1};\qquad \qquad y''(x)=\sum_{n=0}^{\infty}(n+r-1)(n+r)C_n~x^{n+r-2}\qquad .....(3)$$
Substituting $(2)$ and $(3)$ in $(1)$ we get,
$$x~\sum_{n=0}^{\infty}(n+r-1)(n+r)C_n~x^{n+r-2}+\sum_{n=0}^{\infty}(n+r)C_n~x^{n+r-1}-\sum_{n=0}^{\infty}C_n~x^{n+r}=0$$
$$\implies \sum_{n=0}^{\infty}(n+r)^2~C_n~x^{n+r-1}~-~\sum_{n=0}^{\infty}C_n~x^{n+r}=0\qquad .....(4)$$
Lowest power of $~x~$ in equation $(4)$ is $~{r-1}~$, so coefficient of $~x^{r-1}~=0$ gives the  indicial equation $~r^2~=0\implies r=0,~0$
From equation $(4)$ we have the following recursive formula,
$$(n+r+1)^2~C_{n+1}~-~C_{n}=0$$
$$\implies C_{n+1}=\frac{1}{(n+r+1)^2}~C_{n}\qquad ........(5)$$
From $(5)$ we have 
$C_1=\frac{1}{(r+1)^2}~C_{0}$
$C_2=\frac{1}{(r+2)^2}~C_{1}=\frac{1}{(r+1)^2~(r+2)^2}~C_{0}$
$C_3=\frac{1}{(r+3)^2}~C_{2}=\frac{1}{(r+1)^2~(r+2)^2~(r+3)^2}~C_{0}$
$\cdots$
Therefore 
$$y(x)=C_0~x^r \left[1+\frac{1}{(r+1)^2}~x+\frac{1}{(r+1)^2~(r+2)^2}~x^2+\frac{1}{(r+1)^2~(r+2)^2~(r+3)^2}~x^3~+\cdots\right]$$
For $~r=0~$, $$y_1(x)= \left[1+~x~+\frac{x^2}{4}+\frac{x^3}{36}+\cdots\right]$$
$$\implies y_1(x)=\sum_{n=0}^{\infty}\frac{x^n}{(n!)^2}=J_0 (2\sqrt{x})$$
$J_0(X)~$ is the modified Bessel function of first kind and order $~0~$.
The other independent solution of equation $(1)$ is  $$y_2(x)=\left[\frac{\partial y}{\partial r}\right]_{r=0}$$
$$\implies y_2(x)=y_1(x)~\log x~-~\left[2~x+\frac{3}{4}~x^2~+\cdots\right]$$
$$\implies y_2(x)=Y_0 (2\sqrt{x})$$
$Y_0(X)~$ is the modified Bessel function of second kind and order $~0~$.
General solution is $$y(x)=A~y_1(x)~+~B~y_2(x)\qquad \text{where $~A,~B~$are constants.}$$
