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Im trying to solve the following equation

[tex]

y''(x) + \frac{y'(x)}{x} + \frac{y(x)}{x^2} = 0

[/tex]

Then with

[tex]y(x) = \sum_{i=0}^\infty a_i*x^{i+k}[/tex]

[tex]y'(x) = \sum_{i=0}^\infty a_i*(i+k)*x^{i+k-1}[/tex]

[tex]y''(x) = \sum_{i=0}^\infty a_i*(i+k)(i+k-1)*x^{i+k-2}[/tex]

I get

[tex]\sum_{i=0}^\infty a_i*(i+k)(i+k-1)*x^{i+k-2} + \sum_{i=0}^\infty a_i*(i+k)*x^{i+k-2} + \sum_{i=0}^\infty a_i*x^{i+k-2} = 0[/tex]

but how can i get a recurrence relation from this.

I need something like

[tex]a_{i+2} = f(i)*a_i[/tex]

But with only the same i+k-2 in all terms i dont know how to proceed.

[tex]

y''(x) + \frac{y'(x)}{x} + \frac{y(x)}{x^2} = 0

[/tex]

Then with

[tex]y(x) = \sum_{i=0}^\infty a_i*x^{i+k}[/tex]

[tex]y'(x) = \sum_{i=0}^\infty a_i*(i+k)*x^{i+k-1}[/tex]

[tex]y''(x) = \sum_{i=0}^\infty a_i*(i+k)(i+k-1)*x^{i+k-2}[/tex]

I get

[tex]\sum_{i=0}^\infty a_i*(i+k)(i+k-1)*x^{i+k-2} + \sum_{i=0}^\infty a_i*(i+k)*x^{i+k-2} + \sum_{i=0}^\infty a_i*x^{i+k-2} = 0[/tex]

but how can i get a recurrence relation from this.

I need something like

[tex]a_{i+2} = f(i)*a_i[/tex]

But with only the same i+k-2 in all terms i dont know how to proceed.

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