Find Values of Lemda ( eigenvalues ) for which the given problem has a non trivial solution. Also determine the corresponding nontrivial solutions ( eigen function )Solve Question 5Priority Question 5Image Attached.. Satisfiesthe given DE and the I.C's as well as B.Cs. If the expansion (i.e. Fourier serieappearing in the solution) converges to a continuous function with continuesecond order partial derivatives, then the formal solution is an actual solutionHomogeneous and nonhomogeneous B.Cs.Boundary conditions of the formu(0, t)%3D%3DTn = (1 'd)n 'In =are nonhomogeneous. When associated with the heat equation, they implythat the endpoints are kept at nonzero temperature. If the endpoints areinsulated, then the B.Cs become homogeneous.1.2.6 ExercisesFind the values of (eigenvalues) for which the given problem has a non-trivial solution. Also determine the corresponding nontrivial solutions (eigenfunctions).1. y" + Xy = 0, 0 + Uyy = 0Chapter 1. Preliminaries on Problems associated with Differential Equations1415ch PDES%3D4. y" + Ay = 0, 0

y''-λy=0,0<x<π,y0-y'0=0,yπ=0 Case-1 if λ=0, we have y''=0 ⇒yx=A+Bxy'x=B By condition, we have…

Answered: 5. y" = Ay = 0, 0

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5. y" = Ay = 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

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Question

Find Values of Lemda ( eigenvalues ) for which the given problem has a non trivial solution. Also determine the corresponding nontrivial solutions ( eigen function )

Solve Question 5

Priority Question 5

Image Attached..

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).

+ 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)

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