Second order differential with substitution

Question

Hey guys, I was doing this question and am really stuck :/

I got up to taking n as 1 and getting z'=sqrt(y)*y'

Can someone tell me where to go from here?

Edit:

I've done the first part, just not sure how to continue with part (i).

Answer 1

Setting $n = 1$, we get $$z = y \left(\frac{dy}{dx}\right)^2$$ and $$ \frac{dz}{dx} = \sqrt{y}\,\frac{dy}{dx} \Rightarrow z = \frac{2}{3}y^{3/2} + c_1 $$

Setting those equal to each other we get $$ \frac{dy}{dx} = \sqrt{\frac{2}{3}y^{1/2} + \frac{c_1}{y}} $$

You now have an IVP to solve.