Tricky Riccati DE w/ exponential terms: dE/dt = cB_t + bE_t + a(E_t)^2, s.t. B_t = g(1-e^{ht})/(1-ke^{ht})


New member
Mar 3, 2019
Hi Guys,

I just found this forum, and I am hoping that someone here can help me with a very difficult Riccati DE that has me scratching my head. I'm starting to wonder if there is even a closed-form solution to it, or if it requires some type of numerical 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.