# A Difficult Differential Equation

1. Aug 1, 2013

So, as a result of a thought experiment, I've got a differential equation, which I can't solve:

$R r'' \sin \frac{r}{R} - 2 (r')^2 \cos \frac{r}{R} - R^2 \cos \frac{r}{R} \sin^2 \frac{r}{R} = 0$, $R > 0$

To make the matters worse, the function $r(\varphi)$ will probably depend on multiple parameters, because when I put $r << R$, I could approximate the equation:

$r r'' - 2 (r')^2 - r^2 = 0$

which gave solution (mostly by lucky guess):

$r = \frac{a}{\sin \varphi + b \cos \varphi}$, $a\in\mathbb R$, $b\in\mathbb R$

Since I'm used only to the simplest types of differential equations, could you please help me and describe every step :shy:

2. Aug 1, 2013

### SteamKing

Staff Emeritus
You've got a homogeneous, second order, non-linear ODE. A closed form solution will be difficult to come by, but numerical methods of solution should work.

3. Aug 1, 2013

### JJacquelin

Hi !
See attachment :

#### Attached Files:

• ###### ODE.JPG
File size:
41.1 KB
Views:
115
4. Aug 1, 2013

Oh, thank you!

5. Aug 1, 2013

### JJacquelin

Sorry, there is a mistake in the attached document:
From d(rho)/d(phi)=f , in the integral the square root should be at denominator instead of numerator. This changes all what follows and result.
Nevertheless, the method to solve the ODE has been explained.

6. Aug 1, 2013

I've noticed :) I'm currently trying to integrate the changed equation.

7. Aug 2, 2013

### JJacquelin

Below, the corrected attachment :
The result is much simpler.

File size:
41.2 KB
Views:
110