use the methods of frobenius to solve:
2xy"+y'+2y =0
i want to know not just the answer but the steps how to answer it (explanation of the answer)
The charachteristic equation is,
$\displaystyle 2r(r-1) + r+0=0$ thus, $\displaystyle r=0,1/2$.
This means to look for solutions of the form:
$\displaystyle y=\sum_{n=0}^{\infty} a_n x^n$ and $\displaystyle y=\sum_{n=0}^{\infty} a_n x^{n+1/2}$
Can you take it from there?
Given the equation,
$\displaystyle x^2p(x)y''+xq(x)y'+r(x)y=0$
Where $\displaystyle p(x),q(x),r(x)$ are analytic functions* on some open interval $\displaystyle (-R,R)$.
The method of Frobenius says that we can look for a solution of the form,
$\displaystyle y=\sum_{n=0}^{\infty}a_n x^{n+r}$ for some number $\displaystyle r$.
Now to find this $\displaystyle r$ we solve the equation,
$\displaystyle r(r-1)p(0) + rq(0) + r(0) = 0$.
So I told you what $\displaystyle r$ has to be.
*)"Analytic" on $\displaystyle (-R,R)$ means the function can be expanded as its Taylor series about any point, in this case about 0.
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