If $\int {yy''dx} = 3xy$ is it possible to find y'? So one of the people I know on discord gave me this problem to solve for fun.
I've tried using integration by parts on the left side and it didn't do anything.
The integral transforms into the ODE $yy''- (3xy)'=0$ but it's non-linear and I have no idea how to solve that. Would a series solution work?
Notice that I'm looking for y' not y so maybe there's some trick there?
Any help is appreciated. Thank you for reading.
Here, $y'$ and $y''$ denote the first and second order derivatives of $y$ w.r.t. $x$.
 A: Disclaimer: I managed to get a recursive definition of the coefficients of the Maclaurin series, but not a closed form.

From the ODE $$yy'' - (3xy)' = 0$$
This expands to $$yy'' - 3xy' -3y = 0$$
Let $y = \sum_{n=0}^{\infty} a_nx^n$. Then the relation is $$\left(\sum_{n=0}^{\infty} a_nx^n\right)\left(\sum_{n=0}^{\infty} (n+2)(n+1)a_{n+2}x^{n}\right) + \sum_{n=0}^{\infty} (-3na_n-3a_n)x^{n} = 0$$
Expanding this, it becomes
$$\sum_{n=0}^{\infty} x^n \left(-3na_n - 3a_n + \sum_{m=0}^n (m+2)(m+1)a_{m+2}a_{n-m}\right) = 0$$
For it to be $0$, each of the coefficients of $x^n$ must be $0$, so it is $$(-3n-3)a_n + \sum_{m=0}^n (m+2)(m+1)a_{m+2}a_{n-m} = 0$$
for all nonnegative integer $n$. Assume that $y(0) = a_0 \not = 0$ and $y'(0) = a_1$ are fixed at some values. Then solving for $a_{n+2}$, it is $$a_{n+2} = \frac{(3n+3)a_n - \sum_{m=0}^{n-1} (m+2)(m+1)a_{m+2}a_{n-m}}{(n+2)(n+1)a_0}$$
For the first few values, this comes out to $a_2 = \frac{3}{2}$, $a_3 = \frac{a_1(3-a_2)}{3a_0} = \frac{a_1}{2a_0}$, $a_4 = \frac{9a_{2}-2a_{2}a_{2}-6a_{3}a_{1}}{12a_{0}} = \frac{3a_{0}-a_{1}^{2}}{4a_{0}^{2}}$.
A: $$3 x y'+3 y=y y'';\;y(0)=1,y'(0)=0$$
Approximated graph of $y'$

Approximated graph of $y$

A: There is impossible to directly integration of both sides , you have to face directly to solve $yy''-(3xy)'=0$ .
$yy''=3xy'+3y$
$y''=\dfrac{3xy'}{y}+3$
you can consider as the ODE of the type http://science.fire.ustc.edu.cn/download/download1/book%5Cmathematics%5CHandbook%20of%20Exact%20Solutions%20for%20Ordinary%20Differential%20EquationsSecond%20Edition%5Cc2972_fm.pdf#page=416
