18. a. Use the re initial value problem o noituloa srd yllensg oM y(to) = 0, y(to) = 0 (35) y" +y = g(1), is L.0.E me1o9rT sin(1 – s)g(s) ds. (36) y = omodnon pi n b. Use the result of Problem 17 to find the solution of the initial value problem y" +y = g(1), y(0) = yo. y(0) = y- %3D
18. a. Use the re initial value problem o noituloa srd yllensg oM y(to) = 0, y(to) = 0 (35) y" +y = g(1), is L.0.E me1o9rT sin(1 – s)g(s) ds. (36) y = omodnon pi n b. Use the result of Problem 17 to find the solution of the initial value problem y" +y = g(1), y(0) = yo. y(0) = y- %3D
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
18th
![form a fundamental set of solutions of the homogeneousy
22. E
the sol
of the integrals appearing in equation (30). This depends entirely
In
where
an expression for the particular solution Y(t) in terms of a
forcing function, or if
forcing functions. (See Problems 18 to 22.)
a
you
where
the cor
nonhor
probler
the eva
o da
Problems
olow that Y(t) is a solution of the initial value problem
y(to) = 0, y(to) = 0.
both t
functio
proble
the out
In each of Problems 1 through 3, use the method of variation of
parameters to find a particular solution of the given differential
equation. Then check your answer by using the method of
undetermined coefficients.
L[y] = g(t),
y(to) = Yo- Y (to) = y,
(32)
17. Show that the solution of the initial value problem
1. y" - 5y +6y = 4e'
2. y"-y-2y = 4e-
3. 4y" – 4y' + y = 8e'/2
%3D
L[y] = y" + p(t) y + q(t) y = g(1),
3.7
initial value problems W
u(1o) = Yo,
u'(to) = Yo-
(33)
In each of Problems 4 through 9, find the general solution of the given
differential equation. In Problems 8 and 9, g is an arbitrary continuous
function.
L[u] = 0,
v (to) = 0,
%3D
w (34)
One c
L[v] = g(t),
oiter
are w
4. y" + y = 2 tan t, -0<t< a/2
5. y" + 4y = 3 csc(2r),
proce
oscill:
of a sl
physi
0 < t < T/2
6. y" +4y' + 4y = 21-?e-24,
7. 4y"+y = 8 sec(t/2),
t > 0
-T <t < T
nao () 2noitsups ni elengotni sd
oun nmols given by equation (30).
) nosupo lo noiuloe los 18. a. Use the result of Problem 16 to show that the solution esu
8. y" +9y = g(1)
9. y"-7y + 10y = g(t)
In each of Problems 10 through 15, verify that the given functions
yı and y2 satisfy the corresponding homogeneous equation; then
find a particular solution of the given nonhomogeneous equation. In
Problems 14 and 15, g is an arbitrary continuous function.
initial value problem ls nso noitul02 orf
y(to) = 0, y'(to) = 0
point
y" +y = g(t),
(35)
physi
solve
the c
is
1.0.E m9109dT
10. y' – 2y = 41² – 3, `t> 0;
11. ty" – t(t + 2)y' + (t + 2)y = 6r³, t>0; y1(t)= t,
Y2(t) = te'
yı(t) = ²,
-
Y2(t) = 1¬1
y =
sin(t – s)g(s) ds.
(36)
Insbno-bnog auoonoomodnon sd ihi
b. Use the result of Problem 17 to find the solution of the initil
probl
12. (1 – t)y" + tỷ -y = 2(t – 1)²e-, 0<t< 1; _y(t) = e',
y2(t) = t
value problem(0q+
beha
13. xy" + xy' + (x² – 0.25)y = 3x³/2 sin.x, x> 0; y (x) =
x-1/2 sin x, y,(x) = x-!/2 cos x
14. (1– x)y" + xy' – y = g(x), 0 < x < 1; y¡(x) = e*,
y2(x) = x
15. xy" + xy + (x² – 0.25)y = g(x), x> 0; y;(x) = x-1/2 sin x,
y2(x) = x-1/2 cosx
16. By choosing the lower limit of integration in equation (30) in the
text as the initial point to, show that Y(t) becomes
y' +y = g(t),o y(0) = yo, y'(0) = yo-
syste
%3D
19. Use the result of Problem 16 to find the solution of the initial
as sh
(posi
mass
value problem
L[y] = [D² – 21D + (2²+ µ?)]y = g(1), i y(to) = 0, y (t) =0.
%3D
acts
is al
of th
%3D
Note that the roots of the characteristic equation are 1 ± iu.
20. Use the result of Problem 16 to find the solution of the initial
value problem
Hooke
Y(t) =
y1(s) y2(1) – y1(t) y2(s)
%3D
g(s) ds.
L[y] = (D – a}y = g(t), y(t,) = 0, y'(to) = 0,
where a is any real number.
10Robert
mooT vd bodip
Micrograp
his law of
roughly, "](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbf6a5413-a9d8-45a3-95ef-c6a752923ebd%2F624102e5-5ce5-442b-a975-0da4c8857b2b%2Fx42ogj9_processed.jpeg&w=3840&q=75)
Transcribed Image Text:form a fundamental set of solutions of the homogeneousy
22. E
the sol
of the integrals appearing in equation (30). This depends entirely
In
where
an expression for the particular solution Y(t) in terms of a
forcing function, or if
forcing functions. (See Problems 18 to 22.)
a
you
where
the cor
nonhor
probler
the eva
o da
Problems
olow that Y(t) is a solution of the initial value problem
y(to) = 0, y(to) = 0.
both t
functio
proble
the out
In each of Problems 1 through 3, use the method of variation of
parameters to find a particular solution of the given differential
equation. Then check your answer by using the method of
undetermined coefficients.
L[y] = g(t),
y(to) = Yo- Y (to) = y,
(32)
17. Show that the solution of the initial value problem
1. y" - 5y +6y = 4e'
2. y"-y-2y = 4e-
3. 4y" – 4y' + y = 8e'/2
%3D
L[y] = y" + p(t) y + q(t) y = g(1),
3.7
initial value problems W
u(1o) = Yo,
u'(to) = Yo-
(33)
In each of Problems 4 through 9, find the general solution of the given
differential equation. In Problems 8 and 9, g is an arbitrary continuous
function.
L[u] = 0,
v (to) = 0,
%3D
w (34)
One c
L[v] = g(t),
oiter
are w
4. y" + y = 2 tan t, -0<t< a/2
5. y" + 4y = 3 csc(2r),
proce
oscill:
of a sl
physi
0 < t < T/2
6. y" +4y' + 4y = 21-?e-24,
7. 4y"+y = 8 sec(t/2),
t > 0
-T <t < T
nao () 2noitsups ni elengotni sd
oun nmols given by equation (30).
) nosupo lo noiuloe los 18. a. Use the result of Problem 16 to show that the solution esu
8. y" +9y = g(1)
9. y"-7y + 10y = g(t)
In each of Problems 10 through 15, verify that the given functions
yı and y2 satisfy the corresponding homogeneous equation; then
find a particular solution of the given nonhomogeneous equation. In
Problems 14 and 15, g is an arbitrary continuous function.
initial value problem ls nso noitul02 orf
y(to) = 0, y'(to) = 0
point
y" +y = g(t),
(35)
physi
solve
the c
is
1.0.E m9109dT
10. y' – 2y = 41² – 3, `t> 0;
11. ty" – t(t + 2)y' + (t + 2)y = 6r³, t>0; y1(t)= t,
Y2(t) = te'
yı(t) = ²,
-
Y2(t) = 1¬1
y =
sin(t – s)g(s) ds.
(36)
Insbno-bnog auoonoomodnon sd ihi
b. Use the result of Problem 17 to find the solution of the initil
probl
12. (1 – t)y" + tỷ -y = 2(t – 1)²e-, 0<t< 1; _y(t) = e',
y2(t) = t
value problem(0q+
beha
13. xy" + xy' + (x² – 0.25)y = 3x³/2 sin.x, x> 0; y (x) =
x-1/2 sin x, y,(x) = x-!/2 cos x
14. (1– x)y" + xy' – y = g(x), 0 < x < 1; y¡(x) = e*,
y2(x) = x
15. xy" + xy + (x² – 0.25)y = g(x), x> 0; y;(x) = x-1/2 sin x,
y2(x) = x-1/2 cosx
16. By choosing the lower limit of integration in equation (30) in the
text as the initial point to, show that Y(t) becomes
y' +y = g(t),o y(0) = yo, y'(0) = yo-
syste
%3D
19. Use the result of Problem 16 to find the solution of the initial
as sh
(posi
mass
value problem
L[y] = [D² – 21D + (2²+ µ?)]y = g(1), i y(to) = 0, y (t) =0.
%3D
acts
is al
of th
%3D
Note that the roots of the characteristic equation are 1 ± iu.
20. Use the result of Problem 16 to find the solution of the initial
value problem
Hooke
Y(t) =
y1(s) y2(1) – y1(t) y2(s)
%3D
g(s) ds.
L[y] = (D – a}y = g(t), y(t,) = 0, y'(to) = 0,
where a is any real number.
10Robert
mooT vd bodip
Micrograp
his law of
roughly, "
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