Solving Higher Order Differential Equations

Question

I have to solve the following differential equation using the substitution $$u = y'$$

$$yy''+(y')^2 + 1 = 0 $$

But how do I integrate after simplifying the substitution if I cannot separate the variables?

Answer 1

Hint -

Let: $$u(y)=\dfrac{dy(x)}{dx}$$

Then: $$ \dfrac{d^2y}{dx^2}=\dfrac{du(y)}{dx}=\dfrac{du(y)}{dy}\dfrac{dy}{dx}=u'(y)y' $$