# Solving ordinary first order quadratic differential equation system

I have a simple, two object thermodynamic model with radiation and advection. This model consists of two first order quadratic differential equations, what I would like to solve analytically. The equations can be simplified to $${{d}\over{dt}}x=a_{0}x^4+a_{1}y^4+a_{2}x+a_{3}y+a_{4}$$ $${{d}\over{dt}}y=a_{5}x^4+a_{6}y^4+a_{7}x+a_{8}y+a_{9}$$ I'm trying to find the analytic solution because I would like to use the model in an embedded system with limited resources, where solving it numerically is not feasible. So far I have tried to solve this with Maxima without any success. I have also consulted with my college textbooks, but they cover only linear differential equation systems.

Any help, idea or direction on how to solve it would be appreciated. Thanks.

(I have moved this question from MathOverflow, it looks more on-topic here.)

• Note that the equiibrium point is a solution of the system. – AnilB Apr 17 '13 at 15:34

Take $h > 0$ a time step. The approximations for $x_i = x(ih), y_i = y(ih)$ are $$\begin{cases} x_{i+1} = x_i + h (a_0x^4+a_1y^4+a_2x+a_3y+a_4)\\ y_{i+1} = y_i + h (a_5x^4+a_6y^4+a_7x+a_8y+a_9) \end{cases}$$
The computational cost should be minimal and the approximation for small $h$ is fairly accurate. If you find this method insufficiently accurate you can always try better numerical methods.