Mathematics 118A, Winter 1998

Homework Assignment 4, due Friday, February 6

1. Certain solutions to the (scalar) equation

ẋ = \sqrt{x}  (*)

fail to be unique.

(a) Give two solutions which are equal at t = 0, but different for every t > 0.

(b) Revisit the proof of local existence and uniqueness of solutions given in class. How does the right side of (*) fail to satisfy the assumptions made in that proof?

2. Consider the system

ẋ = −y − x \sqrt{x^{2} + y^{2}} exp \sqrt{x^{2} + y^{2}},
ẏ = x − y \sqrt{x^{2} + y^{2}} exp \sqrt{x^{2} + y^{2}}.

(a) Determine the critical points and characterize them by linearization.

(b) Is (0, 0) an attractor?

3. Verhulst, Exercise 3.5. Use Mathematica as much as possible for doing the calculation, and graph the phase space for different values of c to illustrate your argument.

4. Use Mathematica to study the damped oscillator equation

ÿ = −y − k ẏ.

(a) Visualize the behavior of the eigenvalues of the system as a function of k.

(b) Solve the equation and plot the phase portrait near the origin for each of the three distinct cases (a pair of complex conjugate eigenvalues, one real eigenvalue, two distinct real eigenvalues). Do not use the Mathematica functions DSolve or MatrixExponential. Instead, carry out the matrix computations (change of basis!) explicitly. You may use the

functions Eigenvalues, Eigenvectors, and JordanDecomposition—consult the

online documentation for details.