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!
Best Answer
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.