Qualitative Differential equations

[dismo]

Problem:
Let x(t), y(t) e a solution of

dx/dt=y+x^2
dydt=x+y^2

with x(t0) NOT = y(t0)

Show that x(t) NOT = y(t) for all t

Attempt:

I feel like the easiest way to show this would be to show that x=y is an orbit of the system and then simply use the fact that orbits may not cross due to the uniqueness of IVP's at every point in the solution space?

So if I set x=y

dx/dt = y+y^2
dy/dt = y+y^2

This implies
dy/dx = 1

So y=x+c for all t.

Is this a reasonable solution?
Is there anything that needs clearing up?

Thanks guys.

 

[HallsofIvy]

you started by assuming that y= x and showed that y= x+ c. How does that show that y= x is an orbit of the system? It might be better to determine a differential equation for y- x.