Request for help with tricky Riccati DE with exponential terms.

Mar 2019
2
0
Germany
Hi all,

I am developing a model that requires me to derive the solution of a rather tricky Riccati DE, and I am having a devil of a time with it and am beginning to wonder if there is even a closed-form solution. Anyhow, the equation is

\(\displaystyle \frac{dE}{dt} = cB_{t} + bE_{t} + aE_{t}^{2},\)

such that

\(\displaystyle B_{t} = g\left(\frac{1-\exp(ht)}{1-k\exp(ht)}\right)\)

and \(\displaystyle a, b, c, g, h,\) and \(\displaystyle k\) are all constant. The intended integration domain for this function is time, which is positive- so we can assume continuity of the \(\displaystyle cB_{t}\).

I'm not really seeing any clear avenues that go anywhere toward a solution. None of the approaches I'm aware of for Riccati DEs seem to be bearing any fruit at all in terms of progress. I'd be greatly appreciative to anyone who can offer anything useful in terms of a path forward- either with respect to a closed-form solution or a quasi-analytic numerical one.

Thank you in advance.
 

skipjack

Forum Staff
Dec 2006
21,493
2,476
Do you know the values of the constants?