Autonomous polynomial differential equation

[janet123]

Homework Statement

Is it possible to find general solution for the following 3rd degree polynomial differential equation:
dx/dt=-a1*x+a2*x^2+a3*x^3

Homework Equations

The Attempt at a Solution

I understand that its is possible to integrate 1/(-a1*x+a2*x^2+a3*x^3), however, end equation that I get integrating directly has the form like - t=f(x)*ln(g(x)), where both f(x) and g(x) are polynomials. And it is clear that from this it won't be possible to express x as a function of t.
Are there any methods how this differential equation can be solved?
Thank you in advance!

 

[HallsofIvy]

What, exactly, do you mean by solve? The fact is that most first order differential equations cannot be solved in the form x(t)= a function of t. The best you can expect is an implicit formula connecting x and t.

 

[JJacquelin]

I understand that its is possible to integrate 1/(-a1*x+a2*x^2+a3*x^3), however, end equation that I get integrating directly has the form like - t=f(x)*ln(g(x)), where both f(x) and g(x) are polynomials. And it is clear that from this it won't be possible to express x as a function of t.
Are there any methods how this differential equation can be solved?

The solutions of the differentiel equation doesn't depend of the method used to solve it. Any method will lead to the same final result as far as the developments are correct.
So, if a method leads to solutions t=f(x)*ln(g(x)) and if it is impossible to express x as a function of t on an expected form (generaly a combination of a finite number of standard functions), the hitch will be the same for the other methods.

 

[AlephZero]

And it is clear that from this it won't be possible to express x as a function of t.

That statement is true for almost every "random" differential equation you can write down.

Most ODEs don't even have solutions that can be expressed at all using "elementary" functions like logs, trig, etc. At least the solution of your equation can be written in the form t = F(x).

This situation is no different from the fact that you can't express the integral of most functions in a "simple" closed form.

In fact, if a differential equation is "interesting" enough, its solutions may have been given names so they can be written down easily. Often they are named after whoever first studied them - Bessel and Hankel functions, Legendre polynomials, Fresnel integrals, etc.

 

[Redbelly98]

Moderator's note: duplicate threads merged.