Problem

It is given that $v=y^4$ and

$$y^3\frac{\mathrm{d}^2y}{\mathrm{d}x^2}+3y^2\left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)^2+y^3\frac{\mathrm{d}y}{\mathrm{d}x}+y^4=e^{-2x}.$$

(a) Show that

$$\frac{\mathrm{d}^2v}{\mathrm{d}x^2}+\frac{\mathrm{d}v}{\mathrm{d}x}+4v=4e^{-2x}.$$

(b) Find $y$ in terms of $x$, given that, when $x=0$, $y=1$ and $\dfrac{\mathrm{d}y}{\mathrm{d}x}=-\dfrac{3}{8}$.