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fluidistic
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Homework Statement
The Bessel DE of order 0 is [itex]x^2y''+xy'+x^2y=0[/itex]. A solution is [itex]J_0(x)-\left ( \frac{x}{2} \right ) ^2+\frac{1}{4}\left ( \frac{x}{2} \right ) ^4+...[/itex]
Show that there's another solution for [itex]x\neq 0[/itex] that has the form [itex]J_0(x)\ln (|x|)+Ax^2+Bx^4+Cx^6+...[/itex] and find the coefficients A, B and C.
Homework Equations
I checked out in Boas's book check out what was worth [itex]J_0(x)[/itex] and to my surprise [itex]J_0(x)=\sum_{n=0}^{\infty} \frac{(-1)^n}{\Gamma (n+1) \Gamma (n+1)}\left( \frac{x}{2} \right ) ^{2n}[/itex]. Now I'm looking at the problem statement and I'm not really sure how the solution looks. It doesn't look beautiful at all to me.
I've been digging into my professor's notes (unfortunately in Spanish) and there's a theorem that states that if the inditial equation leads to 2 roots such that [itex]\mu _1-\mu _2 \in \mathbb{N}[/itex] then there's another linearly independent solution and he gives the form.
He takes as example precisely the Bessel DE with [itex]p=0[/itex], hence our case.
He says that the theorem leads to a solution of the form (knowing that [itex]J_0(x)[/itex] is a solution): [itex]K_0(x)=J_0(x)\ln (|x|)+\sum _{j=1}^{\infty} \frac{(-1)^{j+1}}{(j!)^2} H_j \left ( \frac{x}{2} \right ) ^{2j}[/itex] where [itex]H_k=\sum _{j=1}^{k} j^{-1}[/itex].
The Attempt at a Solution
If I take for granted his results, I get that another solution is [itex]K_0(x)=J_0(x)\ln (|x|)+\frac{x^2}{4}+\frac{3x^4}{128}+\frac{11}{13824}x^6-...[/itex] where it's very easy to see all the coefficient they ask for.
I wonder if it's right and I'm also not understanding well the point of the problem statement when they give [itex]J_0(x)-\left ( \frac{x}{2} \right ) ^2+\frac{1}{4}\left ( \frac{x}{2} \right ) ^4+...[/itex].
I'd appreciate any comment.