How can I solve this non-linear ODE?

Question

I'm trying to solve the following ODE

$\qquad \dot{x}_t = \frac{e^{-x_{t}} r (\pi + \frac{\lambda \gamma}{\lambda + r} - \frac{c}{\lambda}) - e^{\frac{r}{\lambda} x_t} \left( \frac{c}{\lambda (\pi+\gamma)-c}\right)^{1+\frac{r}{\lambda}} \frac{r\lambda\gamma}{\lambda+r} + r (\pi - 2 c / \lambda) - \frac{cr}{\lambda} e^{x_{t}}}{-e^{-x_t} \frac{(\lambda \pi -c)r}{\lambda (\lambda + r)} - e^{\frac{r}{\lambda} x_{t}} \left( \frac{c}{\lambda (\pi+\gamma)-c}\right)^{\frac{r}{\lambda}} \left( \left( \frac{c}{\lambda (\pi+\gamma)-c}\right) \frac{r\lambda \gamma}{\lambda + r} + \frac{rc}{\lambda} \left( \frac{1}{\lambda + r} + \frac{1}{r}\right) \right) +\frac{c}{\lambda} } $

where all the constants are positive real numbers. I tried DSolve but it did not work. For example, naming the constants a to f:

DSolve[
  {y'[x] == 
    (a/E^y[x] - E^((r/l) y[x]) b + c - d E^y[x])/(-(f/E^y[x]) - E^((r/l) y[x]) g + h), 
   y[0] == p}, 
  y[x], x]

But nothing happens (the evaluation runs forever). Can anybody help me to get an analytical solution?

Answer 1

This becomes an integration problem. The generated integrals do not have closed form solution. Maple gives two integrals. I tried to solve one of them to start with, and none of Mathematica, Rubi, Maple nor FriCAS could solve them. FriCAS reported a "Poterntial Pole" on it. So I do not think there is closed form solution to the ODE you have.

I will show what I did, and you are free to duplicate it and see for yourself.

First converted the ODE to Maple, and Maple gave solution in terms of integrals

restart; 
with(MmaTranslator);
ode:=FromMma(`y'[x] == 
    (a/E^y[x] - E^((r/l) y[x]) b + c - d E^y[x])/(-(f/E^y[x]) - E^((r/l) y[x]) g + h)`);

sol:=dsolve({ode,y(0) = p},y(x)):
DEtools[remove_RootOf](sol);

But these integrals do not have closed form solution. Looking at the second one, copied it to Mathematica and tried

integrand1=(g*Exp[a*(l+r)/l]-h*Exp[a]+f)/(b*Exp[a*(l+r)/l]+d*Exp[2*a]-c*Exp[a]-a)

Integrate[integrand1, {a, 0, p}]

Rubi can do part of it:

ShowSteps=False;
Int[integrand1,a]

The parts with Int stuck to them above, means it can't integrate these terms.

Then I tried FriCAS

setSimplifyDenomsFlag(true)
integrate((g*exp(a*(l+r)/l)-h*exp(a)+f)/(b*exp(a*(l+r)/l)+d*exp(2*a)-c*exp(a)-a),a=0..p)

And it can't do it. and it says "potentialPole"

You might want to ask in the Math forum if someone can solve this analytically.