Second order nonlinear ODE - not autonomous

[Prof_Hydro]

second order nonlinear ODE -- not autonomous

This equation has arisen from a steady state problem of diffusion with nonlinear reaction:

(dependent var=c; independent variable=x)

c'' = ko + k1 c + k2 c^2 + k3 c x

ko, etc are constants.

I can obtain a solution if I drop the last term which involves the product of the dependent and independent variable. Is there any way to get a solution with this term?

 

[jackmell]

Hello Prof. Mind if I write it as:

y''-k_0-(k_1+k_3 x)y-k_2 y^2=0

and if you wish an analytical expression for the solution, I suggest two options unless someone comes up with a better way:

(1) Compute an IVP numerically and fit the data to a high-degree polynomial within your range of acceptable error.

(2) Compute the power series y=\sum_{n=0}^{\infty} a_n x^n

I realize there is a square in the power series approach but I would just take the Cauchy product. Would need to figure out the convergence radius of course which I suspect is no more than one. I recommend this approach for the following reasons: It's not in the Handbook of DEs, not in another book on non-linear ones, and Mathematica's DSolve can't solve it. Of course, that doesn't mean a nice way doesn't exists but rather I'm just out of bullets. :)