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I am currently working on the differential equation: $$ x \frac{d^2y}{dx^2} + \frac{1}{2} \frac{dy}{dx}+ \frac{1}{4} y = 0 $$

During a brief discussion with an expert, I was advised to try expressing the solution in the form $y=f\cdot g$, where $f=x^n$. The suggestion was to rewrite the equation in terms of $g$. However, I am struggling to find relevant literature or references on this method.

I would greatly appreciate any insights, guidance, or references that can help me understand and apply this approach. If anyone is familiar with this method or has encountered similar problems, kindly share your knowledge and recommend any books or resources that may assist in solving such equations.

Thank you in advance for your assistance!

LPR
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  • I suggest that you take a look at https://math.stackexchange.com/questions/1804280/verify-y-xaz-p-leftbxc-right-is-a-solution-to-y-left-frac1-2ax-ri?rq=1. – Gonçalo Jan 04 '24 at 05:18

1 Answers1

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Are you sure the hint wasn't to write $y = g(x^n)$? Plugging that into the differential equation, you get $$ n^2 x^{2n-1} g''(x^n) + (n^2 - n/2) x^{n-1} g'(x^n) + g(x^n)/4 = 0$$ If you take $n = 1/2$, that is $$ \frac{g''(\sqrt{x})}{4} + \frac{g(\sqrt{x})}{4} = 0 $$ which is a differential equation you should know how to solve.

Robert Israel
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  • That appears to be the same hint in different notation (cdot between functions expresses composition, not multiplication) – Ben Voigt Jan 04 '24 at 16:16
  • @BenVoigt Actually $\circ$ is usually used for composition, not $\cdot$. And if $f(x) = x^n$ it's $g \circ f$, not $f \circ g$. – Robert Israel Jan 04 '24 at 20:44
  • @RobertIsrael: Ok, but we can't assume that OP knew how to type ∘, or even noticed the distinction. I think the question means composition. – Ben Voigt Jan 04 '24 at 21:47
  • @RobertIsrael Do you have any references where I can refer to more examples like this? – LPR Jan 05 '24 at 00:21