MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

In some physical problem the following differential equation appears


where the dot denotes derivative with respect to $t$. $x$ is evidently a function of $t$. I'm wondering what the theory of differential equations knows about how to solve these kind of equation. Generic solution for arbitrarily nonlinear functions $F$ and and $f$ is desirable without making any assumptions on theses functions.

The solution can easily be found for $F(x)=x$ and for arbitrary $f(t)$. In principle, one could Taylor expand $F(x)$ in the vicinity of the origin, and do perturbation theory in terms of higher corrections, but that's quite trivial and is \textit{not} the subject of my question.

UPD. By making the substitution $x = u'/u$ one can reduced the above equation to a Riccati type equation provided that $F(x)$ is a quadratic polynomial in $x$.

share|cite|improve this question
What IS the subject of your question? Closed form solutions? Hopeless without restricting F or f in some way. Numerical solutions? Pick your textbook. – Michael Renardy Apr 5 '11 at 22:29
Does this DE appear in the above generality for your physical problem, or do $F$ and $f$ live in a fairly specific family of functions? – Ryan Budney Apr 5 '11 at 23:01
Yes, indeed, $F(x)$ needs to be generic. Last thing I can give up on is $f(t)$ -- let's say it's harmonic, but it does not really make the problem easier. – Peter Apr 5 '11 at 23:13
Surely, if such a general theory does not yet exist, it would be worthwhile to create it: I think this is interesting. – Zen Harper Apr 6 '11 at 8:38
I would be surprised if solutions exist for ARBITRARY choices of $F$ and $f$. You probably need some weak hypothesis on the functions $F$ and $f$, for example that they are Lipschitz continuous. – Tom LaGatta Apr 6 '11 at 21:07
up vote 2 down vote accepted

This is not a complete answer either but at this page you can find many special cases of your equation for which the closed form solution is available.

share|cite|improve this answer
Right, Thanks, I recently realized myself that for some special cases it can be reduced to Riccati equation by substituting $x=u'/u$ for example. – Peter Apr 6 '11 at 20:38

There is no general closed-form solution method for this differential equation. Even in some rather simple special cases (e.g., I believe, $F(x) = x^3$, $f(t) = t$, which is an Abel d.e.), there is no known closed-form solution.

share|cite|improve this answer
Yes, indeed, only $F=x + const$ is the case I know the solution for. But since the $x$ and $t$ dependence is additively separated, I thought one may figure it out somehow... – Peter Apr 5 '11 at 23:15
I imagine, however, that the additive separation of $x$ and $t$ dependencies may simplify any analysis of the qualitative behaviour of the solutions, not that I can offer any concrete advice in that regard. This is an entirely different kettle of fish of course, but it is frequently what is really desired. A closed form solution, if sufficiently complicated, may even get in the way of your qualitative understanding. – Harald Hanche-Olsen Apr 6 '11 at 6:53

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

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