Nonlinear Differential Equation Homework Attempt

[Fizz_Geek]

Homework Statement

r" = 1/r^2

Homework Equations

A friend gave this to me, he was just wondering how we'd approach it.

The Attempt at a Solution

I don't think the equation is linear, so I don't know how to approach it. My friend suggested integrating both sides with respect to r, but we don't know if that's legal because we don't know what r might be a function of.

 

[hunt_mat]

Multiple both sides by r' and that will give you the first integral.

 

[Fizz_Geek]

Multiple both sides by r' and that will give you the first integral.

How's does multiplying by a derivative give me an integral?

 

[hunt_mat]

As
$$r''=\frac{1}{r^{2}}$$
Multiply both sides by r' to obtain:
$$r'r''=\frac{r'}{r^{2}}\Rightarrow\left(\frac{(r')^{2}}{2}\right) '=\left( -\frac{1}{r}\right) '$$
Integrate easily from here.

 

[HallsofIvy]

Did you try it? What is the derivative of $(r')^2$? So what is the integral of $r' r''$? What is the integral of $dr/r^2$?

 

[Fizz_Geek]

To hunt_mat:

That's a very nice trick! I got stuck on the next step, integrating (r')², but I'll try to work that out with my friend before asking again.

Thanks very much!

 

[hunt_mat]

It's a standard trick, become familier with it.