Non-linear 2nd ODE involving squares of derivatives

[iqjump123]

Homework Statement

y''+(1/y)*(y')²=0

Homework Equations

The Attempt at a Solution

This is another problem I am having trouble with. I have done searches around the internet, but seen that all "non linear" ODE of second order involves a non linear form in a non differential term (like y''+xy^2=0, or something like that), instead of the DE term.

Punching through wolfram alpha gave a really simple straightforward answer, so I believe it shouldn't be too hard, as long as I get the general method to solve it.

thanks in advance.

 

[dynamicsolo]

The question to ask might be, "Where have we seen something like this:

y'' + (1/y)*(y')² = 0 ?"

We can see that y can't be zero, so let's multiply through by it to get

y y'' + (y')² = 0 .

Now think about the Product Rule -- what is this the derivative of? You will then be led to a separable differential equation. (And I feel like I'm talking like a fortune cookie...)

 

[ehild]

A different hint, which can be applied to all second-order equations which do not contain the independent variable x explicitly: let be the independent variable y, and denote y'=u. Then

y''=u'=(du/dy)(dy/dx)=u (du/dy),

and you get a first-order equation for u.

ehild