I'm trying to solve the following Ricatti equation:

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

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:


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

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

Any hints would be appreciated

  • $\begingroup$ am i missing something? why is this called a riccati equation? $\endgroup$
    – abel
    Mar 24, 2015 at 11:32
  • $\begingroup$ 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... $\endgroup$
    – Vic B
    Mar 24, 2015 at 12:07

1 Answer 1


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)

enter image description here


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.