Consider the di↵erential equation ˙ x = f(x) for each given function f(x). In each case,
(i) Find all equilibrium solutions of the di↵erential equation in the specified interval.
(ii) Graph f(x) and the phase line.
(iii) Determine the stability of each equilibrium solution.
(a) f(x) = 3 ! x2 in the interval (!1, 1)
(b) f(x) = x2 in the interval (!1, 1)
(c) f(x) = x3 ! x in the interval (!1, 1)
(d) f(x) = cos(x) in the interval (!2⇡, 2⇡)
Consider the model for two populations u and v given by the system of nonlinear di↵erential equations
below (where u and v are both nonnegative.
u˙ = u(12 ! 2u ! 2v)
v˙ = v(2u ! 5)
(a) This system has three equilibrium solutions. One of these is (u, v) = (0, 0). Find the other two. At
one equilibrium v equal to zero and u > 0. At the other equilibrium both u and v are positive.
(b) Viewing this system in the form
u˙ = f(u, v), v˙ = g(u, v),
compute the Jacobian matrix
fu denotes the derivative of f with respect to u and similarly for the other entries of A.
(c) Evaluate the matrix A at the equilibrium solution where both u and v are positive. Calculate the
eigenvalues of A and use this to determine whether this equilibrium solution is stable or unstable.