Methods to solve the system of non-linear differential equations.

Question

I have the following system: $$\begin{cases}\frac{dx}{dt} = (x+6)(x-7)(y-7)\\ \frac{dy}{dt} = (x^2-9)(y+5)\end{cases}$$

I have solved the system using the eigenvalue analysis and using 4th order Runga Kutta to get a numerical solution. Are there any other methods I can use to find the plots and the directions of the trajectories?

Thanks in advance!

Answer 1

You have

$$dx/dt = (x+6)(x-7)(y-7) \text{ and }$$ $$dy/dt = (x^2-9)(y+5).$$

Divide both equations to obtain

$$\dfrac{dy}{dx}=\dfrac{(x^2-9)(y+5)}{(x+6)(x-7)(y-7)}$$ $$\implies \dfrac{y-7}{y+5}dy= \dfrac{x^2-9}{(x+6)(x-7)}dx$$

You can try to solve this differential equation by integration to obtain

$$y(x) = f(x,c).$$