For the homogeneous Fredholm equation (x}=1]sin(x+s)y(s d5 the eigenvalue2 and the corresponding eigen function y(x), involving arbitrary constantsA and B, are-2(b) 2=,y(x)= B(sin x+cosx),2(a) 1=, y(x)= A(sin x-cos x)(c) 1=-4,y(x)= B(sin x- cosx) (d) 1 =,=2,>(x)= A(sin x+cos x)

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Answered: For the homogeneous Fredholm equation…

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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The eigenvalues of the initial value problem y' +Ây=0, y(0)=0, y'(7)=0 are a) , = vn, for n=1,2,3,... b) 1, =n², for n=1,2,3,... c) 2, = 2n, for n=1,2,3,... d) , =, for n=1,2,3,... 2'
The eigenfunctions for y" + 2y' + Ay = y(0) = 1, y(1) = 0 are {e" sin(nπa)O {e2r sin(프플프) 1O n=1 {e-"sin(뿌)EO od In=1 4 II {e-2" sin(nπa)이 {e" sin ()O 2 n=1 .2r -:.. ( po
9 - The eigenvalues An and the eigenfunctions yn for the equation y" + Ay = 0 for the boundary conditions y(0) = 0, y(3) =0 are a) O An NTL , Yn = sin 9. n = 1, 2, 3, ... 3. b) NTL , Yn = sin V3 3 ,n= 1, 2, 3, ... c) , Yn = sin(3nTI), n = 1, 2, 3, ... 3. n2 n2 , Yn = sin(2NAI), n= 1, 2, 3,... d) e) An = n² n², Yn = sin(nTr), n = 1, 2, 3, . .. Boş bırak
П Write the function f(x) = x (1-x) as an eigenfunction ex- pansion of the eigenfunctions corresponding to the Sturm-Liouville prob- lem y" + Xy = 0, 0 < x < y(0) = 0 y (H) = 0 πT
Find the smallest eigen value of (1+x)y" + y' + λy = 0, y(0)=0, y(1)=0
the general solution of the inhomogeneous equation y '' = 9x 2 + 2x -1
a' = with one of the phase plane direction fields below: A B D (The blue lines are the arrow shafts, and the black dots are the arrow tipa.) Note: To solve this problem, you only need to compute eiguvalus. In fact, it is enough to just compute whether the cigenvalues are real or complex and positive or negative, If you use Trace-Determinant plane, give a brief explanation about eigenvalues.
6. (a) Let T : L²[0, 2π] → L²[0, 2ñ] be given by 2π Tf(x) = s cos(xt) f(t) dt. Show that (i) T is self-adjoint. (ii) cos x and sin x are eigenvectors of T.
Find the eigenvalues and eigenfunctions yn(x) for the given boundary-value problem. (Give your answers in terms of n, making sure that each value of n corresponds to a unique eigenvalue.) x²y" + xy' + 2y = 0, y(1)= 0, y(e) = 0 2n = Yn(x) = , n = 1, 2, 3, ... , n = 1, 2, 3,...
When solving the Sturm-Liouville boundary value problem y" - Ay=0, 0
7. Let be the operator on P[r]3 defined by L(p(x)) = xp'(x) +p"(x) (1) Find the matrix A representing with respect to [1, x, x²]. (2) Find the matrix B representing with respect to [1, x, 1 + x²]. (3) Find the matrix S such that B S-¹ AS. (4) If p(x) = ao + a₁x + a₂(1 + x²), = calculate L¹ (p(x)).

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For the homogeneous Fredholm equation y(*)=2] sin(x+5) the'eigenvalue a and the corresponding eigen function y(x), involving arbitrary constants A and B, are 2 -2 (a) 1=,y(x)= A(sin x- cos x) (b) 1=(x)= B(sin x + cos x)., -2 (c) 2=,(x)= B(sin x- cosx) (d) a==,r(x)= A(sin x-+cosx) %3D