2- Solve the problem 0
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- 6. Consider the eigenvalue problem y" + ày = 0; y'(0) = 0, y(1) + y'(1) = 0. All the eigenvalues are nonnegative, so write à = a² where a 2 0. (a) Show that A = 0 is not an eigen- value. (b) Show that y = Acos ax + B sin ax satis- fies the endpoint conditions if and only if B = 0 and a is a positive root of the equation tan z = 1/z. These roots {an}° are the abscissas of the points of intersection of the curves y = tan z and y = 1/z, as indicated in Fig. 3.8.13. Thus the eigenvalues and eigenfunctions of this problem are the numbers {a;}° and the functions {cos an x}9°, re- spectively. y = 2n Зл I ly = tan zIf f is a function then (x)(x-2) is equal to Oa. O a. A2) Ob. A2)(x) Oc A2)8(x+ 2) O f(2)(x-2) O e No correct answerTwo numbers are greater than 10 but less than 40. Their GCF is 17. What are the two numbers?
- 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.Answer either has to be an integer or in decimals1- Solve the problem uz = Uxx + sin(nx) + sin(2nx) fu(0, t) = 0 lu(1,t) = 0 u(x, 0) = 0 P.D.E. 0When solving the Sturm-Liouville boundary value problem y" - Ay=0, 0The eigenvalue problem y"+y = 0, y'(0) = 0, y' P. (7/2) = 0 0 has the solution Select the correct answer. (A) y = sin(2nx), λ = 4n², n = 1,2,3.... B D E y = cos(2nx), λ = 4n², n = 1,2,3,... y = sin(2nx), λ = 2n, n = 1,2,3,... y = cos(2nx), λ = 2n, n = 1,2,3,... none of these+) Solve Poisson problem for a=lı bal ру u(x₁b) = f₂(x) Du= f(x,y) u(0₁9)=9₁ (9) b 0 u(a.y) = 9₂ (y) >x a u(x₁) = f(x) Figure 1. f(x,y) = sin2TX₁ f₁ = £₂=0, 9₁=9₂=0 using one dimensional eigen function expansion.Recommended textbooks for youAdvanced Engineering MathematicsAdvanced MathISBN:9780470458365Author:Erwin KreyszigPublisher:Wiley, John & Sons, Incorporated
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