Use u = y' substitution to solve (y + 1) y'' = (y')^2

[s3a]

Homework Statement

Solve the following differential equation, by using the substitution u = y'.:
(y + 1) y'' = (y')^2

Homework Equations

I'm assuming: Chain Rule

The Attempt at a Solution

My problem is, simply, that I don't get how to go from u = y' to y'' = u du/dy, and I would appreciate it if someone could show me how!

Other than that, I should be fine.

 

[maajdl]

Re-write:

(y+1) u' = u²

then rearrange to get u's on the left and y's on the right and integrate.
(this is a classical manipulation in classical mechanics)

 

[LCKurtz]

Homework Statement

Solve the following differential equation, by using the substitution u = y'.:
(y + 1) y'' = (y')^2

Homework Equations

I'm assuming: Chain Rule

The Attempt at a Solution

My problem is, simply, that I don't get how to go from u = y' to y'' = u du/dy, and I would appreciate it if someone could show me how!

Other than that, I should be fine.

I will use prime for differentiation with respect to the independent variable which I will assume to be $x$. We have the substitution $u(y) = y'$. Differentiating both sides with respect to $x$ gives$$
y'' = u'(y) = \frac{du}{dy}\cdot y' = u \frac{du}{dy}$$It's just the chain rule and I switched the factors in the last step.