6.6 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 Laplace transform of a + b t + c t 2 a b c
Your Turn
Find the Laplace transform of t i f t ≥ 1 , 0 i f t < 1 .
Your Turn
Find the inverse Laplace transform of 8 s 3 ( s + 2 )
Answer
2
t
2
−
2
t
+
1
−
e
−
2
t
Your Turn
Verify Table 6.2.1.
Your Turn
Using the Heaviside function write down the piecewise function that is 0 t < 0 , t 2 t [ 0 , 1 ] t t > 1
Your Turn
Using the Laplace transform solve
m
x
″
+
c
x
′
+
k
x
=
0
,
x
(
0
)
=
a
,
x
′
(
0
)
=
b
.
where m > 0 , c > 0 , k > 0 c 2 − 4 k m > 0
Your Turn
Using the Laplace transform solve
m
x
″
+
c
x
′
+
k
x
=
0
,
x
(
0
)
=
a
,
x
′
(
0
)
=
b
.
where m > 0 , c > 0 , k > 0 c 2 − 4 k m < 0
Your Turn
Using the Laplace transform solve
m
x
″
+
c
x
′
+
k
x
=
0
,
x
(
0
)
=
a
,
x
′
(
0
)
=
b
.
where m > 0 , c > 0 , k > 0 c 2 = 4 k m
Your Turn
Show the differentiation of the transform property. Suppose ℒ { f ( t ) } = F ( s )
ℒ
{
−
t
f
(
t
)
}
=
F
′
(
s
)
.
Hint: Differentiate under the integral sign.
Your Turn
Solve x ‴ + x = t 3 u ( t − 1 ) x ( 0 ) = 1 x ′ ( 0 ) = 0 x ″ ( 0 ) = 0
Your Turn
Let us think of the mass-spring system with a rocket from Example 6.2.2. We noticed that the solution kept oscillating after the rocket stopped running. The amplitude of the oscillation depends on the time that the rocket was fired (for 4 seconds in the example).
Find a formula for the amplitude of the resulting oscillation in terms of the amount of time the rocket is fired. Is there a nonzero time (if so what is it?) for which the rocket fires and the resulting oscillation has amplitude 0 (the mass is not moving)? Your Turn
Define
(
t
−
1
)
2
i
f
1
≤
t
<
2
,
3
−
t
i
f
2
≤
t
<
3
,
0
o
t
h
e
r
w
i
s
e
.
Sketch the graph of f ( t ) Write down f ( t ) Solve x ″ + x = f ( t ) , x ( 0 ) = 0 , x ′ ( 0 ) = 0 Your Turn
Using the Heaviside function u ( t )
0
i
f
t
<
1
,
t
−
1
i
f
1
≤
t
<
2
,
i
f
2
≤
t
.
Answer
f
(
t
)
=
(
t
−
1
)
(
u
(
t
−
1
)
−
u
(
t
−
2
)
)
+
u
(
t
−
2
)
Your Turn
Solve x ″ − x = ( t 2 − 1 ) u ( t − 1 ) x ( 0 ) = 1 , x ′ ( 0 ) = 2
Answer
x
(
t
)
=
(
2
e
t
−
1
−
t
2
−
1
)
u
(
t
−
1
)
−
1
2
e
−
t
+
3
2
e
t
Your Turn
Let f ( t ) = t 2 t ≥ 0 g ( t ) = u ( t − 1 ) f * g
Your Turn
Let f ( t ) = t t ≥ 0 g ( t ) = sin t t ≥ 0 f * g
Your Turn
Find the solution to
m
x
″
+
c
x
′
+
k
x
=
f
(
t
)
,
x
(
0
)
=
0
,
x
′
(
0
)
=
0
,
for an arbitrary function f ( t ) m > 0 , c > 0 , k > 0 c 2 − 4 k m > 0
Your Turn
Find the solution to
m
x
″
+
c
x
′
+
k
x
=
f
(
t
)
,
x
(
0
)
=
0
,
x
′
(
0
)
=
0
,
for an arbitrary function f ( t ) m > 0 , c > 0 , k > 0 c 2 − 4 k m < 0
Your Turn
Find the solution to
m
x
″
+
c
x
′
+
k
x
=
f
(
t
)
,
x
(
0
)
=
0
,
x
′
(
0
)
=
0
,
for an arbitrary function f ( t ) m > 0 , c > 0 , k > 0 c 2 = 4 k m
Your Turn
Write down the solution to x ″ − 2 x = e − t 2 , x ( 0 ) = 0 , x ′ ( 0 ) = 0 e − t 2
Your Turn
Let f ( t ) = cos t t ≥ 0 g ( t ) = e − t f * g
Answer
1
2
(
cos
t
+
sin
t
−
e
−
t
)
Your Turn
Solve x ″ + x = sin t , x ( 0 ) = 0 , x ′ ( 0 ) = 0
Answer
1
2
(
sin
t
−
t
cos
t
)
Your Turn
Solve x ‴ + x ′ = f ( t ) , x ( 0 ) = 0 , x ′ ( 0 ) = 0 , x ″ ( 0 ) = 0
Answer
∫
0
t
f
(
τ
)
(
1
−
cos
(
t
−
τ
)
)
d
τ
Your Turn
Solve (find the impulse response) x ″ + x ′ + x = δ ( t ) , x ( 0 ) = 0 , x ′ ( 0 ) = 0 .
Your Turn
Solve (find the impulse response) x ″ + 2 x ′ + x = δ ( t ) , x ( 0 ) = 0 , x ′ ( 0 ) = 0 .
Your Turn
A pulse can come later and can be bigger. Solve x ″ + 4 x = 4 δ ( t − 1 ) , x ( 0 ) = 0 , x ′ ( 0 ) = 0 .
Your Turn
Suppose that f ( t ) g ( t ) f ( t ) = g ( t ) = 0 t ≤ 0
( f * g ) ′ ( t ) = ( f ′ * g ) ( t ) = ( f * g ′ ) ( t ) .
Your Turn
Suppose that L x = δ ( t ) , x ( 0 ) = 0 , x ′ ( 0 ) = 0 x = e − t t > 0 L x = t 2 , x ( 0 ) = 0 , x ′ ( 0 ) = 0 t > 0
Your Turn
Compute ℒ − 1 { s 2 + s + 1 s 2 }
Your Turn
Suppose we have a beam of length 1 F = 1 x = 3 4 E I = 1 y ( x )
Your Turn
Solve (find the impulse response) x ″ = δ ( t ) , x ( 0 ) = 0 , x ′ ( 0 ) = 0
Answer
x
(
t
)
=
t
Your Turn
Solve (find the impulse response) x ′ + a x = δ ( t ) , x ( 0 ) = 0 , x ′ ( 0 ) = 0
Answer
x
(
t
)
=
e
−
a
t
Your Turn
Suppose that L x = δ ( t ) , x ( 0 ) = 0 , x ′ ( 0 ) = 0 x ( t ) = cos ( t ) t > 0 L x = sin ( t ) , x ( 0 ) = 0 , x ′ ( 0 ) = 0 f o r t > 0
Answer
x
(
t
)
=
(
cos
*
sin
)
(
t
)
=
1
2
t
sin
(
t
)
Your Turn
Compute ℒ − 1 { s 2 s 2 + 1 }
Answer
δ
(
t
)
−
sin
(
t
)
Your Turn
Compute ℒ − 1 { 3 s 2 e − s + 2 s 2 }
Answer
3
δ
(
t
−
1
)
+
2
t
Your Turn
Solve
y t + y x = 1 , 0 < x < ∞ , t > 0 , y ( 0 , t ) = 1 , y ( x , 0 ) = 0 .
Your Turn
Solve
y t + α y x = 0 , 0 < x < ∞ , t > 0 , y ( 0 , t ) = t , y ( x , 0 ) = 0 .
Your Turn
Solve
y t + 2 y x = x + t , 0 < x < ∞ , t > 0 , y ( 0 , t ) = 0 , y ( x , 0 ) = 0 .
Your Turn
For an α > 0
y t + α y x + y = 0 , 0 < x < ∞ , t > 0 , y ( 0 , t ) = sin ( t ) , y ( x , 0 ) = 0 .
Your Turn
Find the corresponding ODE problem for Y ( x ) t
y t t + 3 y x x + y x t + 3 y x + y = sin ( x ) + t , 0 < x < 1 , t > 0 , y ( 0 , t ) = 1 , y ( 1 , t ) = t , y ( x , 0 ) = 1 − x , y t ( x , 0 ) = 1 .
Do not solve the problem.
Your Turn
Write down a solution to
y t = y x x , 0 < x < ∞ , t > 0 , y x ( 0 , t ) = e − t , y ( x , 0 ) = 0 ,
as an definite integral (convolution).
Your Turn
Use the Laplace transform in t
y t t = y x x , − ∞ < x < ∞ , t > 0 , y t ( x , 0 ) = sin ( x ) , y ( x , 0 ) = 0 .
Hint: Note that e s x s → ∞ x e − s x s → ∞ x Your Turn
Solve
y t + y x = 1 , 0 < x < ∞ , t > 0 , y ( 0 , t ) = 0 , y ( x , 0 ) = 0 .
Answer
y
=
(
x
−
t
)
u
(
t
−
x
)
+
t
Your Turn
For a c > 0
y t + y x + c y = 0 , 0 < x < ∞ , t > 0 , y ( 0 , t ) = sin ( t ) , y ( x , 0 ) = 0 .
Answer
y
=
e
−
c
x
sin
(
t
−
x
)
u
(
t
−
x
)
Your Turn
Find the corresponding ODE problem for Y ( x ) t
y t t + 3 y x x + y = x + t , − 1 < x < 1 , t > 0 , y ( − 1 , t ) = 0 , y ( 1 , t ) = 0 , y ( x , 0 ) = ( 1 − x 2 ) , y t ( x , 0 ) = 0 .
Do not solve the problem.
Answer
s
2
Y
(
x
)
−
s
(
1
+
x
2
)
+
3
Y
″
(
x
)
+
Y
(
x
)
=
x
s
+
1
s
2
,
Y
(
−
1
)
=
0
,
Y
(
1
)
=
0
.
Your Turn
Use the Laplace transform in t
y t t = y x x , − ∞ < x < ∞ , t > 0 , y t ( x , 0 ) = x 2 , y ( x , 0 ) = 0 .
Hint: Note that e s x s → ∞ x e − s x s → ∞ x Answer
y
=
t
x
2
+
t
3
3
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 .
