How Do You Solve Coupled Differential Equations with Non-Linear Terms?

[Rotan72]

dy/dt=xy+ay
dx/dt=bx+yx^2

I don't know how to solve the equations because I never took a class in diff equations when I was still in college[/size]

psss Thank you

 

[HallsofIvy]

If you had taken a class in diff equations, you would probably have learned not to try to solve things like that! That looks to me to be a pair of rather nasty non-linear equations. Generally speaking, one can't expect to get a closed form solution for non-linear equations.

I note that there are two equilibrium solutions: x= 0, y= 0, and x= a, y= -b/a are constant solutions.
In the vicinity of (0,0), the equation linearize to dy/dt= ay, dx/dt= bx.
In the vicinity of (a,-b/a), the equations linearize to dy/dt= -bx+ a²y and dx/dt= -bx/a + 2ay.

 

[Tide]

Perhaps you could make more progress with

$$\frac {dy}{dx} = \frac {y (x + a)} {x (b + xy)}$$