# Help with solving an homogeneous second order ODE or a Riccati equation

I'm trying to solve the following Ricatti equation:

$$\frac{dy}{dt}=ay^2+by+\frac{e^{\theta t}}{c+de^{\theta t}}$$ after making the following substitution $y(t)=-\frac{u'(t)}{au(t)}$ and some algebra then I'm left with a nonlinear homogenous ODE $$u''-bu'+\frac{au}{ce^{-\theta t}+d}=0$$

where $a,b,c,d$ and $\theta$ are constants.

I know that for these kind of equations if you have a particular solution $y_p$, you can come up with a general form for the solution. However, I haven't been able to come up with one.

I know there should be an explicit form since when you put the ODE with some values for $a,b,c$ in Wolfram Alpha it gives you an explicit solution but in terms of the hypergeometric function or Meijer G-function which I think is too complicated and should have a simpler form:

https://www.wolframalpha.com/input/?i=y%27%27-2y%27%2B%281%2F%283%2Bexp%28-2t%29%29%29y%3D0

I tried expressing the equation in Sturm Liouville form but don't know what to do after that :S

$$\frac{d}{dt}[e^{-bt}u']+\frac{ae^{-bt}}{(ce^{-\theta t}+d)}u=0$$

Any hints would be appreciated

• am i missing something? why is this called a riccati equation? – abel Mar 24 '15 at 11:32
• True! This is not a Ricatti equation but an ODE that you get from a Ricatti equation, I'll include the Ricatti equation so maybe people have more ideas... – Vic B Mar 24 '15 at 12:07

The general solution of the ODE is rather complicated. The closed form includes Gauss hypergeometric functions (attachement)

Refer to : http://mathworld.wolfram.com/HypergeometricFunction.html , Equation (2) and general solution (9)