For the system of differential equations:
x'(t) = -(9/5)x + (5/3)y + 2xy
y'(t) = -(18/5)x + (20/3)y - xy
the critical point (xâ‚€, yâ‚€) with xâ‚€ > 0, yâ‚€ > 0 is xâ‚€ = , yâ‚€ =
Change variables in the system by letting x(t) = xâ‚€ + u(t), y(t) = yâ‚€ + v(t). The system for u, v is:
u' =
v' =
Use u and v for the two functions, rather than u(t) and v(t).
For the u, v system, the Jacobian matrix at the origin is:
A = [[â—»,], [â—»,]]
For the system of differential equations:
x'(t) = -x + 3y + 2xy
y'(t) = -18x + 2y - xy
the critical point (o with > 0, yâ‚€ > 0 is
Change variables in the system by letting (t) = o + u(t), y(t) = yâ‚€ + v(t). The system for , is:
Use u and v for the two functions, rather than u(t) and v(t).
For the u, system, the Jacobian matrix at the origin is:
A =
You should note that this matrix is the same as J(o, yâ‚€) from the previous problem.
