TITLE: Solve $y''+(1-2x \cos x \cos 2x)y=0$
QUESTION [6 upvotes]: Solve the differential equation
$$y''+(1-2x \cos x \cos 2x)y=0 \space $$

REPLY [3 votes]: Your best choice is to find an approximate solution using power series techniques. Here is few terms of the series with initial conditions $y(0)=1$ and $y'(0)=1.$
$$ y \left( x \right) =1+x-\frac{1}{2}\,{x}^{2}+\frac{1}{6}\,{x}^{3}+{\frac {5}{24}}\,{x}
^{4}-{\frac {37}{120}}\,{x}^{5}.$$ 
However, if you are interested in finding a solution close to the point $x=0$, then you can consider the differential equation
$$ y''(x)+y(x)=0, $$
which is easy to solve. Here are the plots. The first corresponds to the power series solution and the second to the equation $y''(x)+y(x)=0$ with initial conditions $y(0)=1$ and $y'(0)=1$