Solve the Differential Equation : $y'' + \frac{L(x)}{2}y = 0 $ 
Solve the $2$nd order differential equation :
  $$y'' + \frac{L(x)}{2}y = 0  $$ 

I have been looking at different methods as known integral, variation of parameters  etc, but I don't know how to go about this equation.
Please help.
 A: This is indeed a quite difficult problem to proceed with, without knowing $L(x)$, since there isn't a known general formula. 
There is a method that can help with differential equations that have a function of $x$, $f(x)$ as a coefficient of the derivatives of the solutions and it's by using power series to determine the solution of differential equations.
The general idea, is that we're looking for a solution of the form : 
$$y=\sum_{n=0}^\infty c_nx^n = c_0 + c_1x + c_2x^2 + \dots$$
We proceed by substituting this expression into the given differential equation to determine the values of the coefficients $c_0,c_1,c_2,\dots$
You'll have to calculate the derivatives of the series given, so you can plug it in, depending on your given differential equation each time :
$$y'= \sum_{n=1}^\infty nc_nx^{n-1}$$
$$y'' = \sum_{n=2}^{\infty}n(n-1)c_nx^{n-2}$$
$$\dots$$
There are many things to look after in such a method and many variations (finding a solution via power series around a specific point, which some times has a big significance). You can read some more stuff here .
