For the given equation:
$y'' + x^2 y'+(x-1)y = 0$
I tried the Frobenius method but didn't get far. I'm not sure I understood it that well since it's my first time encountering this method. If anyone could explain it in a simpler manner and help me understand through the given problem, that would be really great. Thank you so much in advance!
First of all, you have to understand & note some concepts.
$\textbf{Analytic function :}$ A function $f(x)$ defined on an interval containing the point $x=x_0$ is called $\textit{analytic }$ at $ x_0$ if its Taylor series $\sum_{n=0}^{\infty}\frac{f^{(n)}(x_0)}{n!}(x-x_0)^n$ exists and $\textit{converges}$ to $f(x)$ for all $x$ in the interval of convergence of its Taylor series.
$\textbf{Ordinary Point :}$ A point $x=x_0$ is called an $\textit{ordinary point}$ of the differential equation $\frac{d^2 y}{dx^2}+P(x)\frac{dy}{dx}+Q(x)y=0$ if both the functions $P(x)$ and $Q(x)$ are analytic at $x=x_0$.
$\textbf{Singular Point :}$ If the point $x=x_0$ is not an ordinary point of the differential equation $\frac{d^2 y}{dx^2}+P(x)\frac{dy}{dx}+Q(x)y=0$, then it is called $\textit{singular point}$.
Generally we use Frobenius method only if $x=0$ is not an ordinary point of the given differential equation. In this case, $x=0$ is indeed an ordinary point of the differential equation. So we get power series solution near the ordinary point $x=0$. I will give you the sketch of solution.
First of all, assume that $y=\sum_{n=0}^{\infty}c_nx^n$ is a solution of the given differential equation.
(Here $c_n$'s are unknown. So our main focus is to determine the value of $c_n$ for all $n$.)
After differentiating $y$ with respect to $x$, we get $\frac{dy}{dx}=\sum_{n=1}^{\infty}nc_nx^{n-1}$ and $\frac{d^2 y}{dx^2}=\sum_{n=2}^{\infty}n(n-1)c_nx^{n-2}$. Substitute these values in original differential equation and get $\sum_{n=0}^{\infty}A_nx^n=0$; where each $A_n$ are now some functions of the coefficients $c_0,c_1,c_2,....$
Comparing the both sides, we get each coefficient $A_n=0$. Now obtain recurrence relation, i.e., obtain $c_n$ in the form of $c_0$ and $c_1$ OR only in $c_0$ whatever possible. Put these all values in the original power series $y=\sum_{n=0}^{\infty}c_nx^n$ and you are done.