Does anyone know a solution to this ODE?

[stephanie22]

Dear all,

I am wondering if anyone knows the solution to the following nonlinear ODE,
$$\left( -3 + \frac{ f(r) f'(r)}{r} \right)(1+ f'(r)^2 ) + f(r) f''(r) = 0$$
subject to the initial conditions f(R) = f(-R) = f_0.

I have a feeling a closed form solution exists to this ODE because I know a very simple closed form solution to the related ODE
$$\left( 2 + \frac{ f(r) f'(r)}{r} \right)(1+ f'(r)^2 ) + f(r) f''(r) = 0 \,.$$
A solution to this ODE is
$$f(r) = (R^2 - r^2)^{1/2} \,.$$

Thanks in advance for any help!

Steph

 

[jvicens]

Dear Steph,

Thank you for posting your question. I am a scientist who specializes in solving nonlinear ODEs and I may be able to help you with your problem. The ODE you have presented is a bit complex and it may not have a closed form solution. However, there are a few things we can try to find a solution or at least approximate it.

One approach is to use numerical methods to approximate the solution. This involves discretizing the ODE and solving it using numerical techniques such as Euler's method, Runge-Kutta methods, or finite difference methods. These methods may provide a good approximation to the solution, but they may not give an exact solution.

Another approach is to use perturbation methods. This involves expanding the solution in terms of a small parameter and finding approximate solutions to the ODE. This method is useful when the ODE has a small parameter that can be used to simplify the problem. However, it may not work for all cases and may also result in approximate solutions.

Lastly, you can also try to find symmetry solutions to the ODE. This involves finding solutions that are invariant under certain transformations. These solutions may provide a simpler form of the ODE and can help in finding an exact solution.

I hope these suggestions help in finding a solution to your ODE. If you need more assistance, please don't hesitate to reach out to me. Best of luck!