What is the missing term in the differential equation?

[jacobrhcp]

Given the following differential equation:

\frac{dy}{dx}=\frac{\sigma y(\alpha x^{\alpha-1}y^{\beta}-\delta-\rho)}{x^\alpha y^\beta-\delta x-y}<br />
and starting condition x(0)=x₀ (=3, for instance)

and these parameters \alpha = 0.2; \beta = 0.1; \rho = 0.014; \delta = 0.05; b = 0.5; \sigma = 0.5;

I want to find an exact solution. What I have tried so far is using the Mathematica tool DSolve, but when executed, Mathematica kept running for over 20 minutes without giving an answer.

I don't see any smart substitution or a way to make separation of variables possible. Do you see any? I'm not asking for a complete solution or anything, but if you know a good book or manual where I can learn how to solve this, that's fine too.

Thanks!

 

[Steely Dan]

There is no algorithm for finding the exact solution of a non-linear ODE (and most of them do not have exact solutions). Series solution techniques may be helpful if what you're interested in is an analytical result, but the bottom line is that you are unlikely to be able to find an exact solution in closed form to this equation.

 

[jacobrhcp]

=( that's unfortunate. I really expected there to be one for some reason. Thanks for your help.

 

[AlephZero]

The fact that $d/dx(x^\alpha y^\beta - \delta x) = \alpha x^{\alpha-1}y^\beta - \delta$ has got to be good for something - but it's not obvious (to me) how to use it.

 

[Steely Dan]

Well, if y is a function of x, then you are missing a term in that equation.