3.10 Exercises
These are homework exercises to accompany Libl's "Differential Equations for Engineering " Textmap. This is a textbook targeted for a one semester first course on differential equations, aimed at engineering students. Prerequisite for the course is the basic calculus sequence.
Your Turn
Find the general solution of x 1 ′ = x 2 − x 1 + t , x 2 ′ = x 2
Your Turn
Find the general solution of x 1 ′ = 3 x 1 − x 2 + e t , x 2 ′ = x 1
Your Turn
Write x ″ + y 2 y ′ − x 3 = sin ( t ) , y ″ + ( x ′ + y ′ ) 2 − x = 0
Your Turn
Suppose two masses on carts on frictionless surface are at displacements x 1 x 2 F x 1
Example 1
Suppose the tanks are as in Example 3.1.2, starting both at volume V r 1 r 2
Your Turn
Find the general solution to y 1 ′ = 3 y 1 , y 2 ′ = y 1 + y 2 , y 3 ′ = y 1 + y 3
Answer
y 1 = C 1 e 3 x y 2 = y ( x ) = C 2 e x + C 1 2 e 3 x y 3 = y ( x ) = C 3 e x + C 1 2 e 3 x
Your Turn
Solve y ′ = 2 x , x ′ = x + y , x ( 0 ) = 1 , y ( 0 ) = 3
Answer
x = 5 3 e 2 t − 2 3 e − t y = 5 3 e 2 t + 4 3 e − t
Your Turn
Write y 1 ″ + y 1 + y 2 = t y 2 ″ + y 1 − y 2 = t 2
Answer
y 3 ′ + y 1 + y 2 = t y 4 ′ + y 1 − y 2 = t 2 y 1 ′ = y 3 y 2 ′ = y 4
Your Turn
Suppose two masses on carts on frictionless surface are at displacements x 1 x 2 x 1 ( 0 ) = x 2 ( 0 ) = 0 x 1 ′ ( 0 ) = x 2 ′ ( 0 ) = a a
Answer
x 1 = x 2 = a t
Your Turn
Suppose the tanks are as in Example 3.1.2 except that clean water flows in at the rate s s
Draw the picture. Set up the system of equations. Intuitively, what happens as t Answer
Left to reader x 1 ′ = r V ( x 2 − x 1 ) x 2 ′ = r V x 1 − r − s V x 2 As t x 1 x 2 Your Turn
Compute determinant of [ 1 2 3 1 4 0 5 0 6 0 7 0 8 0 10 1 ]
Your Turn
For which h [ 1 2 3 4 5 6 7 8 h ] h
Your Turn
Find 3 nonzero 2 × 2 A , B , C A B = A C B ≠ C
Your Turn
Solve [ 1 1 1 − 1 ] x → = [ 10 20 ]
Answer
15
−
5
Your Turn
Suppose a , b , c M = [ a 0 0 b ] , N = [ a 0 0 0 b 0 0 0 c ]
Compute M − 1 Compute N − 1 Answer
1
a
0
0
1
b
1
a
0
0
0
1
b
0
0
0
1
c
Your Turn
Verify that the system x → ′ = [ 1 3 3 1 ] x → [ 1 1 ] e 4 t [ 1 − 1 ] e − 2 t Write down the general solution. Write down the general solution in the form x 1 = ? x 2 = ? Your Turn
Verify that [ 1 1 0 ] e t [ 1 − 1 1 ] e t [ 1 − 1 1 ] e 2 t
Your Turn
Take the system x 1 ′ + x 2 ′ = x 1 x 1 ′ − x 2 ′ = x 2
Write it in the form A x → ′ = B x → A B Compute A − 1 x → ′ = P x → Your Turn
Are [ c o s h ( t ) 1 ] [ e t 1 ] [ e − t 1 ]
Answer
No. cosh ( t ) 1
Your Turn
Write x 1 ′ = 2 t x 2 x 2 ′ = 2 t x 2 Solve and write the solution in matrix notation.Add exercises text here. Answer
0
2
t
0
2
t
C
2
e
t
2
+
C
1
C
2
e
t
2
Your Turn
Find the general solution of x 1 ′ = 2 x 1 , x 2 ′ = 3 x 2 x → ′ = A x → Solve the system by solving each equation separately and verify you get the same general solution. Your Turn
Find the general solution of x 1 ′ = 3 x 1 + x 2 , x 2 ′ = 2 x 1 + 4 x 2
Your Turn
Find the general solution of x 1 ′ = x 1 − 2 x 2 , x 2 ′ = 2 x 1 x 2
Your Turn
Compute eigenvalues and eigenvectors of 9 − 2 − 6 − 8 3 6 10 − 2 − 6 Find the general solution of x → ′ = A x → Your Turn
Let a , b , c , d , e , f a b c 0 d e 0 0 f
Your Turn
Compute eigenvalues and eigenvectors of 1 0 3 − 1 0 1 2 0 2 Solve the system x → ′ = A x → Answer
Eigenvalues: 4 , 0 , − 1 1 0 1 0 1 0 3 5 − 2
1
0
1
Your Turn
Compute eigenvalues and eigenvectors of 1 1 − 1 0 Solve the system x → ′ = A x → Answer
Eigenvalues: 1 + 3 i 2 1 − 3 i 2 − 2 1 − 3 i − 2 1 + 3 i
−
2
cos
(
3
t
2
)
cos
(
3
t
2
)
+
3
sin
(
3
t
2
)
Your Turn
Solve x 1 ′ = x 2 , x 2 ′ = x 1
Answer
1
1
Your Turn
Solve x 1 ′ = x 2 , x 2 ′ = − x 1
Answer
cos
(
t
)
−
sin
(
t
)
Your Turn
Take the equation m x ″ + c x ′ + k x = 0 m > 0 , c ≥ 0 , k > 0
Convert this to a system of first order equations. Classify for what m c k Can you explain from physical intuition why you do not get all the different kinds of behavior here? Your Turn
Can you find what happens in the case when P = [ 1 1 0 1 ]
Your Turn
Can you find what happens in the case when P = [ 1 1 1 1 ]
Your Turn
Which behaviors are possible if P a 0 0 b a b
Your Turn
Take the system from Example 3.1.2, x 1 ′ = r V ( x 2 − x 1 ) x 2 ′ = r V ( x 1 − x 2 ) t
Your Turn
Take [ x y ] ′ = [ 0 1 0 0 ] [ x y ]
Answer
The solution does not move anywhere if y = 0 y x y x 0 x = C 1 t + C 2 y = C 1
Your Turn
challenging
Let us take the example in Figure 3.6.3 with the same parameters as before: m 1 = 2 , k 1 = 4 , k 2 = 2 , m 2 cos ( 5 t ) m 2
Note
Note
This idea is called dynamic damping. In practice there will be a small amount of damping and so any transient solution will disappear and after long enough time, the first mass will always come to a stop.
Your Turn
Let us take the example in Figure 3.6.2 with parameters m 1 = m 2 = 1 , k 1 = k 2 = 1
Your Turn
Suppose there are three carts of equal mass m k
Answer
m 0 0 0 m 0 0 0 m
Solution: 1 − 2 1
Your Turn
Suppose a cart of mass 2 kg k = 1 3 kg k = 1 1
Answer
x
2
=
(
2
5
)
cos
(
1
6
t
)
−
(
2
5
)
cos
(
t
)
Your Turn
Let A = [ 5 − 4 4 0 3 0 − 2 4 − 1 ] .
What are the eigenvalues? What is/are the defect(s) of the eigenvalue(s)? Find the general solution of x → ′ = A x → Your Turn
Let A = [ 2 1 0 0 2 0 0 0 2 ]
What are the eigenvalues? What is/are the defect(s) of the eigenvalue(s)? Find the general solution of x → ′ = A x → Your Turn
Let A = [ 0 1 2 − 1 − 2 − 2 − 4 4 7 ]
What are the eigenvalues? What is/are the defect(s) of the eigenvalue(s)? Find the general solution of x → ′ = A x → Your Turn
Let A = [ 0 4 − 2 − 1 − 4 1 0 0 − 2 ]
What are the eigenvalues? What is/are the defect(s) of the eigenvalue(s)? Find the general solution of x → ′ = A x → Your Turn
Let [ 2 1 − 1 − 1 0 2 − 1 − 2 4 ]
What are the eigenvalues? What is/are the defect(s) of the eigenvalue(s)? Find the general solution of x → ′ = A x → Your Turn
Suppose that A is a 2 × 2 λ A = λ I
Your Turn
Let A = [ 1 1 1 1 1 1 1 1 1 ]
What are the eigenvalues? What is/are the defect(s) of the eigenvalue(s)? Find the general solution of x → ′ = A x → Answer
3
,
0
,
0
No defects.
1
1
1
Your Turn
Let A = [ 1 3 3 1 1 0 − 1 1 2 ]
What are the eigenvalues? What is/are the defect(s) of the eigenvalue(s)? Find the general solution of x → ′ = A x → Answer
1
,
1
,
2
Eigenvalue 1 1
0
1
−
1
Your Turn
Let A = [ 2 0 0 − 1 − 1 9 0 − 1 5 ]
What are the eigenvalues? What is/are the defect(s) of the eigenvalue(s)? Find the general solution of x → ′ = A x → Answer
2
,
2
,
2
Eigenvalue 2 2
0
3
1
Your Turn
Let A = [ a a b c ] a b c [ 1 0 ] A
Answer
5
5
0
5
Your Turn
Using the matrix exponential, find a fundamental matrix solution for the system, x ′ = 3 x + y , y ′ = x + 3 y
Your Turn
Find a fundamental matrix solution for the system, x 1 ′ = 7 x 1 + 4 x 2 + 12 x 3 , x 2 ′ = x 1 + 2 x 2 + x 3 , x 3 ′ = − 3 x 1 − 2 x 2 − 5 x 3 x → = [ 0 1 − 2 ]
Your Turn
(challenging)
Suppose A B = B A e A + B = e A e B
Your Turn
Use Exercise 3.8 .5 ( e A ) − 1 = e − A e A
Your Turn
Suppose A − 1 1 [ 1 1 ] [ 0 1 ]
Find matrix A Find the fundamental matrix solution to x → ′ = A x → Solve the system in with initial conditions x → ( 0 ) = [ 2 3 ] Your Turn
Suppose that A n × n λ A = λ 𝐼
Your Turn
Let A = [ − 1 − 1 1 − 3 ]
Find e t A Solve x → ′ = A x → x → ( 0 ) = [ 1 − 2 ] Your Turn
For any positive integer n A n
3
0
0
9
5
2
4
7
0
1
0
0
2
1
0
2
Your Turn
Compute e t A A = [ 1 − 2 − 2 1 ]
Answer
e
3
t
+
e
−
t
2
e
−
t
−
e
3
t
2
e
−
t
−
e
3
t
2
e
3
t
+
e
−
t
2
Your Turn
Compute e t A A = [ 1 − 3 2 − 2 1 2 − 1 − 3 4 ]
Answer
2
e
3
t
−
4
e
2
t
+
3
e
t
3
e
t
2
−
3
e
3
t
2
−
e
3
t
+
4
e
2
t
−
3
e
t
2
e
t
−
2
e
2
t
e
t
2
e
2
t
−
2
e
t
2
e
3
t
−
5
e
2
t
+
3
e
t
3
e
t
2
−
3
e
3
t
2
−
e
3
t
+
5
e
2
t
−
3
e
t
Your Turn
Compute e t A A = [ 3 − 1 1 1 ] Solve x → ′ = A x → x → ( 0 ) = [ 1 2 ] Answer
(
t
+
1
)
e
2
t
−
t
e
2
t
t
e
2
t
(
1
−
t
)
e
2
t
(
1
−
t
)
e
2
t
(
2
−
t
)
e
2
t
Your Turn
Compute the first 3 terms (up to the second degree) of the Taylor expansion of e t A A = [ 2 3 2 2 ] e 0.1 A
Answer
1 + 2 t + 5 t 2 3 t + 6 t 2 2 t + 4 t 2 1 + 2 t + 5 t 2 1.25 0.36 0.24 1.25
Your Turn
For any positive integer n A n
7
4
−
5
−
2
−
3
4
−
6
−
7
0
1
1
0
Answer
5
(
3
n
)
−
2
n
+
2
4
(
3
n
)
−
2
n
+
2
5
(
2
n
)
−
5
(
3
n
)
5
(
2
n
)
−
4
(
3
n
)
3
−
2
(
3
n
)
2
(
3
n
)
−
2
3
−
3
n
+
1
3
n
+
1
−
2
1 0 0 1 n 0 1 1 0 n Your Turn
Find a particular solution to x ′ = x + 2 y + 2 t , y ′ = 3 x + 2 y − 4
using integrating factor method, using eigenvector decomposition, using undetermined coefficients. Your Turn
Find the general solution to x ′ = 4 x + y − 1 , y ′ = x + 4 y − e t
using integrating factor method, using eigenvector decomposition, using undetermined coefficients. Your Turn
Find the general solution to x 1 ′ ′ = − 6 x 1 + 3 x 2 + cos ( t ) , x 2 ″ = 2 x 1 − 7 x 2 + 3 cos ( t )
using eigenvector decomposition, using undetermined coefficients. Your Turn
Find the general solution to x 1 ″ = − 6 x 1 + 3 x 2 + cos ( 2 t ) , x 2 ″ = 2 x 1 − 7 x 2 + cos ( 2 t )
using eigenvector decomposition, using undetermined coefficients. Your Turn
Take the equation
1
t
−
1
1
1
t
Check that t sin t − t cos t
is the complementary solution. Use variation of parameters to find a particular solution. Your Turn
Find a particular solution to x ′ = 5 x + 4 y + t , y ′ = x + 8 y − t
using integrating factor method, using eigenvector decomposition, using undetermined coefficients. Answer
The general solution is (particular solutions should agree with one of these): x ( t ) = C 1 e 9 t + 4 C 2 e 4 t − t 3 − 5 54 y ( t ) = C 1 e 9 t − C 2 e 4 t + t 6 + 7 216
Your Turn
Find a particular solution to x ′ = y + e t , y ′ = x + e t
using integrating factor method, using eigenvector decomposition, using undetermined coefficients. Answer
The general solution is (particular solutions should agree with one of these): x ( t ) = C 1 e t + C 2 e − t + t e t y ( t ) = C 1 e t − C 2 e − t + t e t
Your Turn
Solve x 1 ′ = x 2 + t , x 2 ′ = x 1 + t x 1 ( 0 ) = 1 , x 2 ( 0 ) = 2
Answer
1
1
Your Turn
Solve x 1 ″ = − 3 x 1 + x 2 + t , x 2 ″ = 9 x 1 + 5 x 2 + cos ( t ) x 1 ( 0 ) = 0 , x 2 ( 0 ) = 0 , x 1 ′ ( 0 ) = 0 , x 2 ′ ( 0 ) = 0
Answer
1
9
Adapted from Differential Equations for Engineers by Jiří Lebl (Oklahoma State University), hosted on LibreTexts (math.libretexts.org) and licensed under CC BY-SA 4.0. Changes were made. License: CC-BY-SA-4.0 .
