Solving differential equations using surface intersections

[marellasunny]

Take for example a system of differential equations:
dx/dt=f(x,y)
dy/dt=g(x,y)

If I plot the surfaces for f(x,y) and g(x,y) and find their intersections,what does this symbolize in terms of the differential equations?
I'm thinking equilibrium points(??)
Can one go further and find the solutions to the differential equations from this?

 

[Simon Bridge]

If I plot the surfaces for f(x,y) and g(x,y) and find their intersections,what does this symbolize in terms of the differential equations?

The intersection between the surfaces would be the set of points in x-y space which satisfy f(x,y)=g(x,y) ... which would mean that dx/dt=dy/dt=v(t) ... i.e. the x component of the velocity is equal to the y-component.
What does that suggest to you?

 

[marellasunny]

The intersection between the surfaces would be the set of points in x-y space which satisfy f(x,y)=g(x,y) ... which would mean that dx/dt=dy/dt=v(t) ... i.e. the x component of the velocity is equal to the y-component.
What does that suggest to you?

Well,in projectile motions where the throw angle is 45 deg,when the 2 velocity components are equal,this implies,the initial velocity in x-dir=inital velocity in y-dir.
But,for other throw angles,I do not know how knowing the point where the 2 velocity components are equal would help.

In terms of growth rate,it could also mean a optimal point of our system.

In turbine flow velocity diagrams,this means the absolute velocity=√2 *relative velocity between water and blades.

Q.But,again coming back to my question,can finding this surface intersections have any significance when it comes to finding the solution to the system of differential equations?A google search gives me a article on "lie groups and surface intersections".

 

[Simon Bridge]

You've pretty much answered your own question - the intersections would be useful in special problems like finding a particular kind of optimal point for the system. As a special trick for solving DEs ... it does not really tell you much about the general solution to the system of DEs unless you have reason to believe that f(x,y)=g(x,y). But that would be a boundary condition in the statement of the problem anyway wouldn't it?