MATH 215/255
Fall 2014
Assignment 7
due 11/12
§6.3, §6.4, §3.1, §3.3
Solutions to selected exercises can be found in [Lebl], starting from page 303.
• 6.3.3: Find the solution to
mx00 + cx0 + kx = f (t), x(0) = 0, x0 (0) = 0
for an arbitrary function f (t), where m > 0, c > 0, k > 0, and c2 − 4km > 0 (system
is overdamped). Write the solution as a definite integral.
• 6.3.4: Find the solution to
mx00 + cx0 + kx = f (t), x(0) = 0, x0 (0) = 0
for an arbitrary function f (t), where m > 0, c > 0, k > 0, and c2 − 4km < 0 (system
is underdamped). Write the solution as a definite integral.
• 6.3.104: Solve x000 + x0 = f (t), x(0) = 0, x0 (0) = 0, x00 (0) = 0 using convolution.
Write the solution as a definite integral.
• 6.4.1: Solve (find the impulse response) x00 + x0 + x = δ(t), x(0) = 0, x0 (0) = 0.
• 6.4.6: Compute L−1 { s
2 +s+1
s2
}.
• 6.4.103: Suppose that L(x) = δ(t), x(0) = 0, x0 (0) = 0, has the solution x(t) = cos(t)
for t > 0. Find (in closed form) the solution to L(x) = sin(t), x(0) = 0, x0 (0) = 0 for
t > 0.
• 3.1.2: Find the general solution of x01 = x2 − x1 + t, x02 = x2 .
• 3.1.4: Write ay 00 + by 0 + cy = f (x) as a first order system of ODEs.
• 3.1.102: Solve y 0 = 2x, x0 = x + y, x(0) = 1, y(0) = 3.
• 3.3.1: Write the system x01 = 2x1 − 3tx2 + sin t, x02 = et x1 + 3x2 + cos t in the vector
form ~x0 = P (t)~x + f~(t), where P (t) is a 2 × 2 matrix.
• 3.3.2: Consider linear system
0
~x =
1 3
3 1
~x.
a) Verify that the system has the following two solutions
1
1
~x(t) = e−2t
, ~x(t) = e4t
.
−1
1
b) Show that they are linearly independent and write down the general solution.
c) Write down the general solution in the form x1 (t) =?, x2 (t) = ? (i.e. write down a
formula for each element of the solution).
• 3.3.104: a) Write x01 = 2tx2 , x02 = 2tx2 in matrix notation. b) Solve and write the
solution in matrix notation.