Question

(a) Find the equilibrium solution, or critical point, of the given system.

(b) Use a computer to draw a direction field and phase portrait centered at the critical point.

(c) Describe how solutions of the system behave in the vicinity of the critical point.

x′ =−0.25x−0.75y+8, y′ =0.5x+y−11.5

(d) Let x= x_c+u and y= y_c+v, where x_c and y_c give the critical point you found in (a). Plug these into the system and show that you obtain a homogeneous system u′ = Au for u = (u v)^T .

(e) Solve the resulting homogeneous system for u and v, and show that the solutions you obtain match the phase portrait that you generated in (b).

Answer 1

the direction field is:

The phase portrait is :