Trying to get exact solution for a nonlinear DE

[Paalfaal]

Ok, this should be quite simple. I've been looking at this problem for quite some time now, and I'm tired.. Please help me!

The equation to solve is $r'$ = $r(1-r^2)$

The obvious thing to do is to do partial fraction expansion and integrate(from r$_{0}$ to r):

$∫ (\frac{1}{r} + \frac{1}{1-r^2})dr = t$

After some trig substitution: $ln[r\sqrt{\frac{r+1}{1-r}}] = t$ (evaluated from r$_{0}$ to r). Then take the exponential on both sides.

$r\sqrt{\frac{r+1}{1-r}} = r_{0}\sqrt{\frac{r_{0}+1}{1-r_{0}}}e^{t}$

This leads to kind of a nasty expression which can't be solved explicitly for $r(t)$ (at least it seems that it can't be solved explicitly).. So this is where I'm stuck.

The solution to this ploblem is: $r(t) = \frac{r_{0}e^{t}}{\sqrt{1+r_{0}(e^{2t}-1)}}$

Again, this should be fairly simple, but I'm stuck.. Please help me!

 

[JJacquelin]

$∫ (\frac{1}{r} + \frac{1}{1-r^2})dr = t$ is false. r is missing on the second fraction.

 

[Paalfaal]

Thank you! Got it now :)

 

[HallsofIvy]

Ok, this should be quite simple. I've been looking at this problem for quite some time now, and I'm tired.. Please help me!

The equation to solve is $r'$ = $r(1-r^2)$

The obvious thing to do is to do partial fraction expansion and integrate(from r$_{0}$ to r):

$∫ (\frac{1}{r} + \frac{1}{1-r^2})dr = t$

That's an incorrect "partial fraction expansion". $1- x^2= (1- x)(1+ x)$ so
$$\frac{1}{r(1- r^2)}= \frac{1}{r}- \frac{1}{r- 1}- \frac{1}{r+ 1}$$
There is no "trig substitution" involved in integrating that. the integral is
$$ln(r)- ln(r-1)- ln(r+1)+ C= ln\left(\frac{r}{r^2-1}\right)+ C$$

After some trig substitution: $ln[r\sqrt{\frac{r+1}{1-r}}] = t$ (evaluated from r$_{0}$ to r). Then take the exponential on both sides.

$r\sqrt{\frac{r+1}{1-r}} = r_{0}\sqrt{\frac{r_{0}+1}{1-r_{0}}}e^{t}$

This leads to kind of a nasty expression which can't be solved explicitly for $r(t)$ (at least it seems that it can't be solved explicitly).. So this is where I'm stuck.

The solution to this ploblem is: $r(t) = \frac{r_{0}e^{t}}{\sqrt{1+r_{0}(e^{2t}-1)}}$

Again, this should be fairly simple, but I'm stuck.. Please help me!

You have integrated incorrectly.

 

[blue_raver22]

I can understand your frustration with trying to find an exact solution for a nonlinear differential equation. This is a common challenge in mathematics and science, as many real-world problems cannot be solved analytically. However, there are various numerical methods that can be used to approximate solutions to nonlinear DEs, such as Euler's method or Runge-Kutta methods. These methods involve breaking down the problem into smaller, simpler steps and using iterative calculations to approximate the solution. It may not give an exact solution, but it can provide a close enough approximation for practical purposes. I would suggest consulting with a mathematician or using computer software to help with these calculations. Keep in mind that sometimes, the best solution is to accept an approximate solution rather than trying to find an exact one.