The differential equation

$\frac{d y}{d x}+x \sin 2 y=x^3 \cos ^2 y$

when transformed to linear form becomes

$\frac{d z}{d x}+2 x z=x^3$

We have,

$\frac{d y}{d x}+x \sin 2 y=x^3 \cos ^2 y$

$\Rightarrow \sec ^2 y \frac{d y}{d x}+(2 \tan y) x=x^3$

Putting tan $y=2$ and $\sec ^2 y \frac{d y}{d x}=\frac{d z}{d x}$, we get

$\frac{d z}{d x}+2 x z=x^3$