(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Use variation of parameters to find the general solution of:

[tex] x^2 y'' + xy' + 9y = -tan(3ln(x)) [/tex]

2. Relevant equations

[tex]y_p = y_2 (x) \int \frac{y_1 (x) g(x)}{W[y_1 (x),y_2 (x)]} dx - y_1 (x) \int \frac{y_2 (x) g(x)}{W[y_1 (x),y_2 (x)]} dx [/tex]

3. The attempt at a solution

Solution for the homogeneous Euler equation is given by

[tex]

y_1 (x) = cos(3ln(x)), y_2 (x) = sin(3ln(x))

[/tex]

I try using these solutions in the equation for variation of parameters with:

[tex] W[y_1 , y_2] = \frac{3}{x} [/tex]

[tex] g(x) = -tan(3ln(x)) [/tex]

The first term I can evaluate, but the second term gives me:

[tex] y_1 (x) \int \frac{x sin^2 (3ln(x))}{3 cos(3ln(x))} dx [/tex]

which doesn't seem solvable by any methods I know...

Any suggestions on what to do from here, or where I went wrong on the way would be helpful.

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# Ordinary Differential Equations

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