MATH 215/255
Homework 7
Submit online via Canvas by 6pm, Thursday, March 17, 2022.
This homework set has written problems only.
§6.3, §6.4, §3.1, §3.3
The written part counts for 20 points.
1. 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.
Hint. Write ms2 + cs + k = m[(s + p)2 + ω12 ] where p =
c
2m ,
ω12 =
k
m
− p2 =
4km−c2
4m2
> 0.
2. Solve x00 + 4x0 + 5x = δ(t − π), x(0) = 0, x0 (0) = 2.
3. Solve x00 + 4x0 + 3x = 2δ(t − π), x(0) = 2, x0 (0) = 0.
4. 3.1.2: Find the general solution of x01 = x2 − x1 + t, x02 = x2 .
5. Rewrite the damped mass-spring system for two masses and three springs
m1 x001 + cx01 = −k1 x1 + k2 (x2 − x1 ),
m2 x002 + cx02 = −k2 (x2 − x1 ) − k3 x2
as a first order system of ODEs. The constants c, mi and ki are positive.
6. 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.
7. 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
−2t
4t
~x(t) = e
, ~x(t) = e
.
−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).
8. 3.3.104: a) Write x01 = 2tx2 , x02 = 2tx2 in matrix notation. b) Solve its general solution, and
write it in matrix notation.
1