A Difficult Differential Equation

[Radek Vavra]

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:

 

[SteamKing]

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.

 

[JJacquelin]

Hi !
See attachment :

 

[Radek Vavra]

Hi !
See attachment :

Oh, thank you!

 

[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.

 

[Radek Vavra]

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.

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

 

[JJacquelin]

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