Most efficient way to solve initial value problem with $(y')^2$?

Question

$y'' + 2x(y')^2 = 0$

$y(0) = 1$

$y'(0) = 4$

I tried the laplace transformation and other methods without success, because this is the first time I deal with $(y')^2$. Just would like to know which method or trick should I use to deal with the $(y')^2$?

Answer 1

Let y' = t.Now, we have,$t' + 2xt^2 = 0$

$\Longrightarrow \frac{-t'}{t^2} = 2x$

$\Longrightarrow\frac{1}{t} = x^2 + C$

We know that,t(0) = 4, so,

C = $\frac{1}{4}$

Put t = y',

$y' = \frac{4}{4x^2+1}$

Integrate both sides and get the result.