4. y" + Xy = 0, 0
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
Find Values of Lemda ( eigenvalues ) for which the given problem has a non trivial solution. Also determine the corresponding nontrivial solutions ( eigen function )
Question 4 Image Attached..
![+ Uyy = 0
Chapter 1. Preliminaries on Problems associated with Differential Equations
14
15
ch PDES
%3D
4. y" + Ay = 0, 0<x < m/2, y'(0) = 0, y'(T/2) = 0.
%3D
pace coordi-
DES, as far
a associated
both initial
of each type
%3D
5. y" = Ay = 0, 0<x < T, y(0) – y'(0) = 0, y(T) = 0.
%3D
%3D
6. y" – 2y + Ay = 0, 0<x < m, y(0) = 0, y(T) = 0.
%3D
7. Discuss why the following PDES can not be solved by the method of sepa-
ration of variables.
space vari-
tion), there
(), where
(as e.g. in
che number
J.r.t. space
(b)
Ugx + Uxy + Uyy = 0
%3D
(a) uz + Uy = 1
8. Show that the PDE
Upp + (1/r) Ur +(1/r²) uoo
with u(r) = R(r) P(0) yields
nditions is
g²R" + rR' – XR = 0, P" + AP = 0
%3D
where A is a constant.
ch satisfies
urier series
continuous
al solution.
9. Show that the PDE
,2
Utt + uz + u = a?u%z
with u(x, t) = X (x)T(t) yields
X" – \X = 0, T" + AT' + (1 – da²)T = 0
%3D
where A is a constant.
10. Show that the PDE
hey imply
points are
Uų = B(uxz + Uyy) with u(x, y, t) = X(x) Y (y) T(t) yields
T' - BuT = 0, X" – XX = 0
%3D
%3D
and
Y" + (A – µ)Y = 0
%3D
where A, u are constants.
.as a non-
11. Show that the PDE
ons (eigen
Upr + (1/r) ur + (1/r2) ug2 + Uzz = 0
with
/r,0, z) = R(r) P(0) Z(z)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F61ff9f11-32a6-4f27-919e-e16ea3be7fb3%2F7a15451b-d64d-46a4-b41a-596a1bedb513%2Fnspzbjt_processed.jpeg&w=3840&q=75)
Transcribed Image Text:+ Uyy = 0
Chapter 1. Preliminaries on Problems associated with Differential Equations
14
15
ch PDES
%3D
4. y" + Ay = 0, 0<x < m/2, y'(0) = 0, y'(T/2) = 0.
%3D
pace coordi-
DES, as far
a associated
both initial
of each type
%3D
5. y" = Ay = 0, 0<x < T, y(0) – y'(0) = 0, y(T) = 0.
%3D
%3D
6. y" – 2y + Ay = 0, 0<x < m, y(0) = 0, y(T) = 0.
%3D
7. Discuss why the following PDES can not be solved by the method of sepa-
ration of variables.
space vari-
tion), there
(), where
(as e.g. in
che number
J.r.t. space
(b)
Ugx + Uxy + Uyy = 0
%3D
(a) uz + Uy = 1
8. Show that the PDE
Upp + (1/r) Ur +(1/r²) uoo
with u(r) = R(r) P(0) yields
nditions is
g²R" + rR' – XR = 0, P" + AP = 0
%3D
where A is a constant.
ch satisfies
urier series
continuous
al solution.
9. Show that the PDE
,2
Utt + uz + u = a?u%z
with u(x, t) = X (x)T(t) yields
X" – \X = 0, T" + AT' + (1 – da²)T = 0
%3D
where A is a constant.
10. Show that the PDE
hey imply
points are
Uų = B(uxz + Uyy) with u(x, y, t) = X(x) Y (y) T(t) yields
T' - BuT = 0, X" – XX = 0
%3D
%3D
and
Y" + (A – µ)Y = 0
%3D
where A, u are constants.
.as a non-
11. Show that the PDE
ons (eigen
Upr + (1/r) ur + (1/r2) ug2 + Uzz = 0
with
/r,0, z) = R(r) P(0) Z(z)
![Satisfies
the given DE and the I.C's as well as B.Cs. If the expansion (i.e. Fourier serie
appearing in the solution) converges to a continuous function with continue
second order partial derivatives, then the formal solution is an actual solution
Homogeneous and nonhomogeneous B.Cs.
Boundary conditions of the form
u(0, t)
%3D
%3D
Tn = (1 'd)n 'In =
are nonhomogeneous. When associated with the heat equation, they imply
that the endpoints are kept at nonzero temperature. If the endpoints are
insulated, then the B.Cs become homogeneous.
1.2.6 Exercises
Find the values of (eigenvalues) for which the given problem has a non-
trivial solution. Also determine the corresponding nontrivial solutions (eigen
functions).
1. y" + Xy = 0, 0<x < T, y(0) = 0, y'(T) = 0.
||
%3D
%3D
3. y" + Ay = 0, 0<x< 2m, y(0) = y(27), y'(0) = y'(2m).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F61ff9f11-32a6-4f27-919e-e16ea3be7fb3%2F7a15451b-d64d-46a4-b41a-596a1bedb513%2Fsyk5fnd_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Satisfies
the given DE and the I.C's as well as B.Cs. If the expansion (i.e. Fourier serie
appearing in the solution) converges to a continuous function with continue
second order partial derivatives, then the formal solution is an actual solution
Homogeneous and nonhomogeneous B.Cs.
Boundary conditions of the form
u(0, t)
%3D
%3D
Tn = (1 'd)n 'In =
are nonhomogeneous. When associated with the heat equation, they imply
that the endpoints are kept at nonzero temperature. If the endpoints are
insulated, then the B.Cs become homogeneous.
1.2.6 Exercises
Find the values of (eigenvalues) for which the given problem has a non-
trivial solution. Also determine the corresponding nontrivial solutions (eigen
functions).
1. y" + Xy = 0, 0<x < T, y(0) = 0, y'(T) = 0.
||
%3D
%3D
3. y" + Ay = 0, 0<x< 2m, y(0) = y(27), y'(0) = y'(2m).
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