Since $x=0$ is singular point: $y(x)=\sum_{n=0}^\infty a_nx^{n+r}$

$$\implies \sum_{n=0}^\infty a_n(n+r)(n+r-1)x^{n+r-1}+\sum_{n=0}^\infty a_nx^{n+r}=0$$

How to proceed further?


1 Answer 1


Rewrite the first series so that the powers match the other:

$$\sum_{n=0}^\infty a_n(n+r)(n+r-1)x^{n+r-1}=a_0r(r-1)x^{r-1}+\sum_{n=0}^\infty a_{n+1}(n+r+1)(n+r)x^{n+r}$$

so that we get


implying that $r=0$ (appears twice modulo 1) so that we will need to use this solution to form a second solution, and that we have



which gives a solution in terms of the Bessel function of the first kind. Since the root is repeated, we then consider

$$y_2=Cy_1\ln(x)+\sum_{n=0}^\infty b_nx^{n+r_2}$$


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