NeeD HELP! Systems of ODE's

[jonscilz]

Does anybody know how to find the critical points of a system of differential equations? I have an exam tonight and I cant seem to understand how my professor does this. Here is an example problem:

Locate all critical points for the system

dx/dt = .004x(50 - x - .75y)
dy/dt = .001y(100 - y - 3.0x)

Hopefully someone here can help me! Thanks!

 

[Guest]

dx/dt = .004x(50 - x - .75y)
dy/dt = .001y(100 - y - 3.0x)

The critical points of a system normally refer to the turning points(max or min points) and the x & y axis intercepts.

The turning points occur when the dx/dt or dy/dt =0

The x intercept occurs when y= 0, y intercept occurs when x=0 (or whatever variables you are using for the 2 axis)
Sub in the 0 value and solve the other unknown gives you a point (this is to be done with the original equation, not the differential form).

Are you sure you have the equations correctly copied as you have 2 variables x & y in the first eqn -should only be 1. Same comment for the other eqn.

Comment please.

 

[jonscilz]

yea im 100% sure... this is how the problem is written in an example sheet i was given by my professor... ill show an example when i get time how the critical points were found for another problem... any more ideas until then?

 

[Unco]

The critical points are where, for the function f(x), f'(x)=0 or f(x) is not differentiable. Note that this does not include axis intercepts (necessarily).

 

[jonscilz]

so can someone show me how this is to be solved?

 

[jonscilz]

this is what I have in my notes regarding critical points of a system of differential equations:

dx/dt = -ax + bxy
dy/dt = -cxy + Dy

Where are dx/dt and dy/dt = 0 simultaneously

-ax + bxy = 0 } x(-a + by) = 0
-cxy + Dy = 0 } y(-cx + D) = 0
x = y = 0 } (0,0) , x = D/C , y = a/b } (D/c , a/b) <--- Critical Value

Example- a=.1 b=.002 c=.0025 D=.2
(80,50) is a critical point


Now I understand the notes but my main question is, does this apply to all systems that are in the form:
dx/dt = -ax + bxy
dy/dt = -cxy + Dy
???????????????????????

either way this doesnt matter because the system has an extra x and y in them which throws off the form as was previously stated? any ideas anybody?

 

[jonscilz]

The critical points are where, for the function f(x), f'(x)=0 or f(x) is not differentiable. Note that this does not include axis intercepts (necessarily).

where are these systems not differentiable?