Which method should I use to solve this second order differential equation?

Question

So I have $$x\frac{d^2y}{dx}+\frac{dy}{dx}+xya^2=0$$ where, $$y=y(x)$$

What I did was to just find the roots using the quadractic formula on its auxiliary equation.

I got $$y(x)=Ae^{x(\frac{-1+\sqrt{1-4x^2a^2}}{2x})}+Be^{x(\frac{-1-\sqrt{1-4x^2a^2}}{2x})}$$

Is this correct or should I use the method of separation of variables? If so, how do I do that?

$\begingroup$ math.stackexchange.com/questions/3173388/… $\endgroup$
– Vadim Chernetsov
Commented Dec 7, 2022 at 9:48 — Vadim Chernetsov, Commented Dec 7, 2022 at 9:48

PierreCarre · Accepted Answer · 2022-12-07 09:55:18Z

0

Your approach is wrong. The method you are trying to use is only valid for linear equations with constant coefficients.

answered Dec 7, 2022 at 9:55

PierreCarre

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$\begingroup$ Which method should I use? $\endgroup$
– mEXsACHINE
Commented Dec 7, 2022 at 9:57
$\begingroup$ The solution is a linear combination of Bessel functions of first and second kind... You will not be able to find a closed form solution that does not use special functions. Can you post the exact origin of the question? $\endgroup$
– PierreCarre
Commented Dec 7, 2022 at 10:15
$\begingroup$ Here's the original question math.stackexchange.com/questions/4593476/… $\endgroup$
– mEXsACHINE
Commented Dec 7, 2022 at 10:29

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