<s>[INST] Find the solutions of : $(x^2-2)x^2y''-(x^2+2)xy'+(x^2+2)y=0$ Find the solutions of :
$(x^2-2)x^2y''-(x^2+2)xy'+(x^2+2)y=0$
I have to find infinite series form solutions,$y=\sum a_nx^{n+r}.$
I got the indicial equation : $r^2-1=0 \implies r_1=-1, r_2=1$
I got the general formula while $r=1$ is $a_{k+2}=\frac{k^2}{2(k+2)^2+4(k+2)}a_k$
Then , one solution is $y_2(x)=x^1\sum a_kx^k$ when I get $a_k$ using $a_{k+2}=\frac{k^2}{2(k+2)^2+4(k+2)} , k>0$.
How can I found the other solution when $r=-1$ ?
Thanks !
 [/INST] The last two terms combine to have a factor $(xy'-y)$. Having this in an equation gives you $y_1=x$ as basis solution. Check your recursion equation, you should get the same, for instance $k=0$ should give $a_2=0$ and then $a_{2k}=0$.
Find the second one by reduction of order as $y_2=u(x)y_1(x)=xu(x)$.
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