Consider the nonlinear system of differential equations dxdt

Consider the nonlinear system of differential equations dx/dt = x(-2x -y + 40) dy/dt = y(-x - y + 30) First find all equilibrium points for this system. Then draw the nullclines for this system. Indicate which are the x- and y- nullclines. Sketch the direction field in the regions between the nullclines. Finally, sketch the solution curves in the phase plane for this system.

Solution

1. To find equilibrium points substitute dx/dt=0 and dy/dt=0

-2x-y+40=0; -x-y+30=0

y=40-2x

on substituting this in 2nd equation

-x-40+2x+30=0

x-10=0

x=10

on substituting this in first equation we have

y=40-2(10)

=20

hence (10,20) is the equilibrium point for the system.

2. The nullclines for the system can be drawn from the equilibrium points

(10,20)

3. To obtain nullclines we have to substitute X\'=0 and Y\'=0

then we have x(-2x-y+40) =0 and (-x-y+30)y=0

the intersection between these two nullclines are equilibrium points

4. To sketch the direction field in the region between the nullclines just take some values of x and y

on taking x i.e., ranging from -5 and 5

x=-5,-4,-3,-2,-1,0,1,2,3,4,5

on substituting these values in the differential equation 1 we can draw the direction field

as well as on taking y from -5 and 5

y=-5,-4,-3,-2,-1,0,1,2,3,4,5

on substituting these values in the differential equation 2 we can draw th direction field

5.From the direction field

from the given equations sketch dy/dt and dx/dt on taking dy/dt= constant for about 10 different choices of the constant after you sketch curve then draw tiny line segments with slope=constant through the curv. Repeat the process.

then check the behaviour of the solution curves from this.