What is the solution to $y*y’’=\sin(x)$? I saw this differential equation in a Khan Academy video. I’m still in high school and I have just began to learn a lot about differential equations, so I wanted to see if i could solve this. I don’t really know how to though. Apologies if it’s a stupid question, that’s just me. 
The differential equation is $$y\frac{d^2y}{dx^2}=\sin(x)$$. If anybody can tell me the answer to this, and how to get it, if it is solvable, I’d love to know. 
Link to the video, the uploader mentions it just at the end but doesn’t really say much about it: https://youtu.be/-_POEWfygmU
 A: While an analytic expression is not feasible for general nonlinear differential equations, one can still apply many of the standard techniques used to solve linear differential equations to obtain approximate expressions. E.g., it's usually quite easy to obtain a solution in the form of a series expansions. In this case, we can try a series expansion around $x = 0$, but we have to note that $\sin(x)$ is an odd function while the product $y y''$ cannot be odd if it is analytic in a neighborhood of $x = 0$.
We can therefore try to find a solution of the form $y(x) = x^{\alpha} y_{\text{an}}(x)$, where $ y_{\text{an}}(x)$ is analytic. This then leads to a series expansion of the form:
$$y(x) = \frac{2}{\sqrt{3}}x^{\frac{3}{2}}\left[1 + a_2 x^2 + a_4 x^4 +\cdots\right]$$
The coefficients are then easily evaluated by inserting this in the differential equation and equating equal powers of $x$, the first few series coefficients are given by:
$$
\begin{split}
a_2 &= -\frac{1}{76}\\
a_4 &= \frac{547}{2945760}\\
a_6 &=-\frac{19079}{14104298880}
\end{split}
 $$
