5.4 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 eigenvalues and eigenfunctions of
y
″
+
λ
y
=
0
,
y
(
0
)
−
y
′
(
0
)
=
0
,
y
(
1
)
=
0
.
Your Turn
Expand the function f ( x ) = x 0 ≤ x ≤ 1
y
″
+
λ
y
=
0
,
y
′
(
0
)
=
0
,
y
(
1
)
=
0
.
Your Turn
Suppose that you had a Sturm-Liouville problem on the interval [ 0 , 1 ] y n ( x ) = sin ( γ n x ) γ > 0 f ( x ) = x , 0 < x < 1
Your Turn
Find eigenvalues and eigenfunctions of
y
′
(
4
)
+
λ
y
=
0
,
y
(
0
)
=
0
,
y
′
(
0
)
=
0
,
y
(
1
)
=
0
y
′
(
1
)
=
0
.
This problem is not a Sturm-Liouville problem, but the idea is the same.
Your Turn
(more challenging)
Find eigenvalues and eigenfunctions for
d
d
x
(
e
x
y
′
)
+
λ
e
x
y
=
0
,
y
(
0
)
=
0
,
y
(
1
)
=
0
.
Hint: First write the system as a constant coefficient system to find general solutions. Do note that Theorem 5.1.1 guarantees λ ≥ 0
Your Turn
Find eigenvalues and eigenfunctions of
y
″
+
λ
y
=
0
,
y
(
−
1
)
=
0
,
y
(
1
)
=
0
.
Answer
λ n = ( 2 n − 1 ) π 2 , n = 1 , 2 , 3 , ⋯ , y n = cos ( ( 2 n − 1 ) π 2 x )
Your Turn
Put the following problems into the standard form for Sturm-Liouville problems, that is, find p ( x ) , q ( x ) , r ( x ) , α 1 , α , β 1 , β 1 ,
x y ″ + λ y = 0 0 < x < 1 , y ( 0 ) = 0 , y ( 1 ) = 0 , ( 1 + x 2 ) y ″ + 2 x y ′ + ( λ − x 2 ) y = 0 − 1 < x < 1 , y ( − 1 ) = 0 , y ( 1 ) + y ′ ( 1 ) = 0 Answer
p ( x ) = 1 , q ( x ) = 0 , r ( x ) = 1 x , α 1 = 1 , α 2 = 0 , β 1 = 1 , β 2 = 0 p ( x ) = 1 + x 2 , q ( x ) = x 2 , r ( x ) = 1 , α 1 = 1 , α 2 = 0 , β 1 = 1 , β 2 = 1 Your Turn
Suppose you have a beam of length 5 y x ( 0 < x < 5 ) y t t + 4 y x x x x = 0 x ( 5 − x ) 2 x y
Your Turn
Suppose you have a beam of length 5 x = 5 u x ( 0 < x < 5 ) u t t = 4 u x x x − 5 50 − ( x − 5 ) 100 u
Your Turn
Suppose the beam is L
Your Turn
Suppose you have
a
4
y
x
x
x
x
+
y
t
t
=
0
(
0
<
x
<
1
,
t
>
0
)
,
y
(
0
,
t
)
=
y
x
x
(
0
,
t
)
=
0
,
y
(
1
,
t
)
=
y
x
x
(
1
,
t
)
=
0
,
y
(
x
,
0
)
=
f
(
x
)
,
y
t
(
x
,
0
)
=
g
(
x
)
.
That is, you have also an initial velocity. Find a series solution. Hint: Use the same idea as we did for the wave equation.
Your Turn
Suppose you have a beam of length 1 y x 0 < x < 1 y t t + 4 y x x x x = 0 sin ( π x ) 0 y
Answer
y
(
x
,
t
)
=
sin
(
π
x
)
cos
(
2
π
2
t
)
Your Turn
Suppose you have a beam of length 10 y x 0 < x < 10 y t t + 9 y x x x x = 0 sin ( π x ) x ( 10 − x )
Answer
9 y x x x x + y t t = 0 ( 0 < x < 10 , t > 0 ) y ( 0 , t ) = y x ( 0 , t ) = 0 y ( 10 , t ) = y x ( 10 , t ) = 0 y ( x , 0 ) = sin ( π x ) y t ( x , 0 ) = x ( 10 − x )
Your Turn
Suppose that the forcing function for the vibrating string is F 0 sin ( ω t ) y p
Your Turn
Take the forced vibrating string. Suppose that L = 1 , a = 1 1 0 < x < 1 − 1 − 1 < x < 0 5.3 .1
Your Turn
The units are cgs (centimeters-grams-seconds). For k = 0.005 , ω = 1.991 × 10 − 7 , A 0 = 20 ± 10
Your Turn
Take the forced vibrating string. Suppose that L = 1 , a = 1 | x | − 1 2 − 1 < x < 1
Answer
y
p
(
x
,
t
)
=
∑
n
odd
n
=
1
∞
−
4
n
4
π
4
(
cos
(
n
π
x
)
−
cos
(
n
π
)
−
1
sin
(
n
π
)
sin
(
n
π
x
)
−
1
)
cos
(
n
π
t
)
.
Your Turn
The units are cgs (centimeters-grams-seconds). For k = 0.01 , ω = 1.991 × 10 − 7 , A 0 = 25
Answer
Approximately 1991 centimeters
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 .
