## Related questions with answers

Question

For each of the following differential equations in Problem, apply the method of Frobenius to obtain the solution. $x^{2} y^{\prime \prime}+\left(x^{2}-x\right) y^{\prime}+2 y=0$

Solution

VerifiedStep 1

1 of 5This differential equation can be written as

$\begin{align*}L[y]=y''+\dfrac{x-1}{x}y'+\dfrac{2}{x^2}y=0\end{align*}$

where

$\begin{align*}&p(x)=\dfrac{x-1}{x}\\&q(x)=\dfrac{2}{x^2}\end{align*}$

which are both fail to be analytic at zero.

However

$\begin{align*}&xp(x)=x-1\\&x^2q(x)=2\end{align*}$

are analytic at zero, we have that $x=0$ is regular singular point.

Now, we are going to find solution in the form

$\begin{align*}&y=\sum_{n=0}^{\infty}c_nx^{n+r}\\& y'=\sum_{n=0}^{\infty}c_n(n+r)x^{n+r-1}\\& y''=\sum_{n=0}^{\infty}c_n(n+r)(n+r-1)x^{n+r-2}\end{align*}$

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