Math 119A, HW 4
Due Oct. 27 in class
1. Use the Laplace transform to solve the initial value problems. Express the solution in
terms of a convolution integral.
(a) y 00 + 2y = g(t);
y(0) = 0, y 0 (0) = 1
(b) y 00 + 4y 0 + 4y = g(t);
y(0) = 2, y 0 (0) = 3
Math 119A, HW 8
Due Dec 1, 2015
1. Verify that (0, 0, 0) is a critical point of each of the following systems and determine
whether it is hyperbolic or nonhyperbolic. If it is hyperbolic, determine whether it is
asymptotically stable or unstable.
(a) x0 =
Math 119A, HW 8
Due 11/24/2015
1. Solve the following initial value problems using the Laplace transform:
0 1 0
sin t
0
t
0
0
0 1 x+
e
(a) x =
, x(0) = 1
1 1 0
1
1
(b) y 00 z = sin t
z 00 + y 0 = cos t
y(0) = 1, y 0 (0) = 0, z(0) = 1, z 0 (0) = 1
2. For
Math 119A, HW 2
Due Oct. 13 in class
1. Use the Laplace transform to solve the initial value problems:
(a) y 00 y 0 6y = 0,
(b) y (4) y = 0,
y(0) = 1, y 0 (0) = 1
y(0) = 1, y 0 (0) = 0, y 00 (0) = 1, y 000 (0) = 0
(c) y 00 2y 0 + 2y = cos t,
y(0) = 1, y 0
Math 119A, HW 10
1. For each of the following systems: (a)Find an equation of the form H(x, y) = c
satisfied by the trjectories. (b)Plot several level curves of the function H(You may
use any computer software). These are trajectories of the given system.
Math 119A, HW 7
Due Nov.17 in class
1. If A =
0 1
cos t sin t
At
, show e =
.
1 0
sin t cos t
Hint: The technique of diagonalization still works if the matrix has complex eigenvalues.
5 3 2
2. Find the general solution of x0 = Ax, where A = 8 5 4.
4 3
3
Math 119A, HW 6
Due Nov.10 in class
1. Let A =
5 6
3 4
(a) Evaluate eAt , where t is any real number.
(b) Find the solution of the initial value problem x0 = Ax,
2 1
0
2. Let A = 0 2 1
0
0 2
x(0) = 2 1
T
(a) Evaluate eAt , where t is any real number(Hint
Math 119A, HW 1
Due Oct. 6 in class
1. Find the Laplace transform of the following functions:
(a) f (t) = t5 et ;
(b) f (t) = t cos 3t;
(c) f (t) = t2 cos 3t;
(d) f (t) = 2t4 + 5 cos 4t;
(e) f (t) = 3e7t 5e2t
2. If Lcfw_f (t) = F (s), show Lcfw_f (at) = a