Solutions 4.2-Page 282
Problem 1
Use Laplace transforms to solve the initial value problems.
x + 4 x = 0 ; x(0) = 5 , x (0) = 0
The necessary transforms are:
L cfw_x(t ) = X ( s )
L cfw_x (t ) = s 2 L cfw_x(t ) s x(0) x (0) = s 2 X ( s ) 5s
L cfw_0 = 0
Su
Solutions 1.5-Page 51
Problem 5
Find general solutions of the differential equations. If an initial condition is given, find
the corresponding particular solution. Primes denote derivatives with respect to x.
xy + 2 y = 3x , y (1) = 5
The differential equ
Solutions 4.6-Page 322
Problem 14
Verify that u (t a) = (t a ) by solving the problem
x = (t a) ; x(0) = 0
to obtain x(t ) = u (t a)
Using direct substitution, u (t a) = x . So we need to show that x(t ) = u (t a ) .
The Laplace transform of the different
Solutions 4.6-Page 322
Problem 3
Solve the initial value problem.
x + 4 x + 4 x = 1 + (t 2) ; x(0) = x (0) = 0
The Laplace transform of the differential equation is:
L cfw_x + L cfw_4 x + L cfw_4 x = L cfw_ + L cfw_ (t 2)
1
s 2 L cfw_x sx (0) x (0) + 4
Problem 21
Find the Laplace transforms of the given functions.
f (t ) = t if t 1; f (t ) = 2 t if 1 t 2; f (t ) = 0 if t > 2
Rewriting f (t ) using the step function yields
f (t ) = t [1 u (t 1)] + (2 t )[u (t 1) u (t 2)] = t tu1 (t ) + 2u1 (t ) tu1 (t )
Solutions 4.5-Page 311
Problem 3
Find the inverse Laplace transform f (t ) the function. Then sketch the graph of f .
F (s) =
es
s+2
Theorem 1 states that L 1 cfw_e as F ( s ) = u (t a ) f (t a) . For this problem,
e s
1
. Therefore L 1
= u (t 1) f (t
Solutions 4.4-Page 299
Problem 16
Apply either Theorem 2 or Theorem 3 to find the Laplace transform of f (t ) .
f (t ) = t 2 cos 2t
Eq.8 (Theorem 2) states that L cfw_t n f (t ) = (1) n F ( n ) ( s ) . Substituting the given f (t ) into
Eq.8 yields:
cfw_
Solutions 4.4-Page 299
Problem 3
Find the convolution f (t ) g (t ) .
f (t ) = g (t ) = sin t
t
Eq.3 states that ( f g )(t ) = f ( ) g (t )d . Direct substitution into Eq.3 yields:
0
( f g )(t ) = sin (sin(t )d . To evaluate the integral, the trigonometri
Solutions 4.4-Page 299
Problem 3
Find the convolution f (t ) g (t ) .
f (t ) = g (t ) = sin t
t
Eq.3 states that ( f g )(t ) = f ( ) g (t )d . Direct substitution into Eq.3 yields:
0
( f g )(t ) = sin (sin(t )d . To evaluate the integral, the trigonometri
Solutions 4.3-Page 292
Problem 13
Use partial fractions to find the inverse Laplace transforms of the functions.
F ( s) =
5 2s
s + 7 s + 10
2
F ( s ) can be rewritten as F ( s ) =
Laplace transform.
A
B
5 2s
=
+
( s + 5)( s + 2) s + 5 s + 2
A( s + 2) + B
Solutions 1.5-Page 51
Problem 27
Solve the differential equation by regarding y as the independent variable rather than x.
( x + ye y )
dy
=1
dx
The differential equation does not yet follow the general form given on pg.43. The
dy
derivative term ( ) is d
Solutions 1.7-Page 79
Problem 3
Separate the variables and use partial fractions to solve the initial value problems.
dx
= 4 x(7 x) , x(0) = 11
dt
Separating variables gives
dx
= dt . Factoring using partial fractions yields:
4 x (7 x )
1
1
dx
28 x + 28(
Solutions 2.3-Page 131
Problem 1
Find the general solutions of the differential equations.
y 4 y = 0
The first step is to find the roots of the characteristic equation.
r2 4 = 0
(r + 2)(r 2) = 0
r = 2,2
Since the roots are real and distinct, the general s
Solutions 4.1-Page 273
Problem 25
Use the transforms in Fig. 4.1.2 to find the inverse Laplace transforms of the functions.
F ( s) =
1
2
5/ 2
ss
From Fig. 4.1.2,
The form of
2
5/ 2
1
corresponds to f (t ) = 1 .
s
corresponds to f (t ) = t a with a = 3 /
Solutions 4.1-Page 272
Problem 5
Apply the definition in (1) to find directly the Laplace transforms of the functions
described (by formula or graph).
f (t ) = sinh t
The definition in (1) states that L cfw_ f (t ) = e st f (t )dt . Direct substitution y
Section 2.8-Page 189
Problem 3
The eigenvalues in Problem 3 are all non-negative. First determine whether = 0 is an
eigenvalue; then find the positive eigenvalues and associated eigenfunctions.
y + y = 0; y ( ) = 0, y ( ) = 0
When = 0 , then y = 0 . The s
Solutions 2.7-Page 176
Problem 1
In the circuit of Fig. 2.7.7, suppose that L = 5 H, R = 25 , and that the source E of emf
is a battery supplying 100 V to the circuit. Suppose also that the switch has been in
position 1 for a long time, so that steady cur
Solutions 2.6-Page 167
Problem 17
The problem gives the parameters for a forced mass-spring-dashpot system with equation
mx + cx + kx = F0 cos t . Investigate the possibility of practical resonance of this
system. In particular, find the amplitude C ( ) o
Solutions 2.6-Page 167
Problem 4
(Note: the solution guide for the book solves the problem for x(0)=0, not x(0)=25)
Express the solution of the given initial value problem as a sum of two oscillations as in
Eq. (8). Throughout, primes denote derivatives w
Solutions 2.5-Page 156
Problem 3
Find a particular solution y p of the given equation. In all these problems, primes denote
derivatives with respect to x.
y y 6 y = 2 sin 3x
The first two derivatives of sin 3 x are cos 3 x and sin 3 x . f ( x) has a finit
Solutions 2.4-Page 140
Problem 3
A mass of 3 kg is attached to the end of a spring that is stretched 20 cm by a force of
15N. It is set in motion with initial position x0 = 0 and initial velocity v0 = 10 m/s.
Find the amplitude, period, and frequency of t
Solutions 2.3b-Page 131
Problem 46
Find a general solution of y iy + 6 y = 0 .
The characteristic equation is r 2 ir + 6 = 0 . The roots can be found by using the
quadratic equation formula or a calculator. Using the quadratic equation formula, the
roots
Solutions 4.3-Page 291
Problem 1
Apply the translation theorem to find the Laplace transforms of the functions.
f (t ) = t 4 e t
The translation theorem states that L cfw_e at f (t ) = F ( s a ) .
For this problem, f (t ) = t 4 and a = .
Therefore L cfw_e
Solutions 4.2-Page 282
Problem 1
Use Laplace transforms to solve the initial value problems.
x + 4 x = 0 ; x(0) = 5 , x (0) = 0
The necessary transforms are:
L cfw_x(t ) = X ( s )
L cfw_x (t ) = s 2 L cfw_x(t ) s x(0) x (0) = s 2 X ( s ) 5s
L cfw_0 = 0
Su
Solutions 4.1-Page 273
Problem 25
Use the transforms in Fig. 4.1.2 to find the inverse Laplace transforms of the functions.
F ( s) =
1
2
5/ 2
ss
From Fig. 4.1.2,
The form of
2
5/ 2
1
corresponds to f (t ) = 1 .
s
corresponds to f (t ) = t a with a = 3 /
Solutions 2.1-Page 106
Problem 3
A homogeneous second-order linear differential equation, two functions y1 and y 2 , and a
pair of initial conditions are given. First verify that y1 and y 2 are solutions of the
differential equation. Then find a particula
Solutions 2.1-2-Pages 107, 119
Problem 20
Determine whether the pairs of functions are linearly independent or linearly dependent
on the real line.
f ( x) = , g ( x) = cos 2 x + sin 2 x
Recall that cos 2 x + sin 2 x =1. Therefore f ( x) = g ( x) .
The pai
Solutions 1.8-Page 89
Problem 2
Suppose that a body moves through a resisting medium with resistance proportional to its
velocity v , so that dv / dt = kv . (a) Show that its velocity and position at times t are
given by
v(t ) = v 0 e kt
and
x(t ) = x0 +
Solutions 1.7-Page 79
Problem 3
Separate the variables and use partial fractions to solve the initial value problems.
dx
= 4 x(7 x) , x(0) = 11
dt
Separating variables gives
dx
= dt . Factoring using partial fractions yields:
4 x (7 x )
1
1
dx
28 x + 28(
Solutions 1.5-Page 51
Problem 27
Solve the differential equation by regarding y as the independent variable rather than x.
( x + ye y )
dy
=1
dx
The differential equation does not yet follow the general form given on pg.43. The
dy
derivative term ( ) is d