EITHER

Show that the substitution \(v=\frac1y\) reduces the differential equation

\(\frac2{y^3}\left(\frac{dy}{dx}\right)^2-\frac1{y^2}\frac{d^2y}{dx^2}-\frac2{y^2}\frac{dy}{dx}+\frac5y=17+6x-5x^2\)

to the differential equation

\(\frac{d^2v}{dx^2}+2\frac{dv}{dx}+5v=17+6x-5x^2\).

Hence find \(y\) in terms of \(x\), given that when \(x=0\), \(y=\frac12\) and \(\frac{dy}{dx}=-1\).