# Homework Help: Use u = y' substitution to solve (y + 1) y'' = (y')^2

1. Mar 11, 2014

### s3a

1. The problem statement, all variables and given/known data
Solve the following differential equation, by using the substitution u = y'.:
(y + 1) y'' = (y')^2

2. Relevant equations
I'm assuming: Chain Rule

3. 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.

2. Mar 11, 2014

### 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)

3. Mar 11, 2014

### LCKurtz

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.