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Given ODE is: $$y''+Py'+Qy=0$$ It has two solutions $f(x)$ and $xf(x)$.

I need to find the solution to the following non-homogeneous ODE: $$y''+Py'+Qy=f(x)$$ where $f(x)$ on the RHS is the solution of the first ODE.

I need to find its solution. I tried to apply variation of parameters method, but failed to proceed and also found this question similar. But being a beginner who's self-learning ODE, I am stuck. An elaboration of its solution would really help me to understand the working of such questions.

EDIT I reconsidered the variation of parameter method and seemed to find a legit solution! $$y=f(x)\left(\int \frac{-xf(x).f(x) dx}{f(x)^2} +c_1\right) + xf(x)\left( \int \frac{f(x). f(x) dx}{f(x)^2} +c_2\right)$$ P.S. The denominator is the wronskian of $f(x)$ and $xf(x)$ which gives $f(x)^2$

On solving, it gives: $$y=f(x) \cdot \left(\frac {x^2}{2}+c_2 x+ c_1\right)$$ Thanks for the valuable comment!

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  • $\begingroup$ What happened when you applied variation of parameters? That seems like a reasonable way of proceeding. $\endgroup$ Commented Aug 21, 2023 at 13:14
  • $\begingroup$ @CW279 thanks for hinting... I completed the solution! Please do suggest if there exists alternative approaches to this! $\endgroup$ Commented Aug 21, 2023 at 13:38
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    $\begingroup$ Solution looks good! That's the answer I got too. Nice work! $\endgroup$ Commented Aug 21, 2023 at 13:44
  • $\begingroup$ @LutzLehmann I edited a typo. Is it good now? $\endgroup$ Commented Aug 21, 2023 at 14:10
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    $\begingroup$ @S.S Just so this doesn't go on the list of answered questions, you might consider typing up your solution and posting it as a proper answer, which you can then accept. $\endgroup$ Commented Aug 21, 2023 at 17:03

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By the method of variation of parameter, one gets the following: $$y=f(x)\left(\int \frac{-xf(x).f(x) dx}{f(x)^2} +c_1\right) + x f(x)\left( \int \frac{f(x). f(x) dx}{f(x)^2} +c_2\right)$$ P.S. The denominator is the wronskian of $f(x)$ and $xf(x)$ which gives $f(x) ^2$

On solving, it gives: $$y=f(x) \cdot \left(\frac {x^2}{2}+c_2 x+ c_1\right)$$

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