Solve the following Sturm-Liouville problem by finding the positive eigenvalues and its corresponding eigenfunctions: y" + ày = 0, y'(0) = 0, y'(1/2) = 0.
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A: x^2 y"(x) + x y'(x)+λ y(x) =0 y'(1)=0 y'(e)=0
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- Solve the following eigenvalue problem: a'' + 2a' + a + λa = 0 given the initial conditions: a'(0) = 0 and a'(1) = 0Find the only real eigenvalue of the problem x²y" - Xxy' + Xy y(1) = 0, y(2) — y' (2) - = 0, 1 < x <2 = 0, and find the corresponding eigenfunction.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 z
- Consider the Sturm-Liouville problem: y" + Ay /(-L) 0, -Lÿ' = -[3. a. Find the eigenvalues and eigenvectors for the coefficient matrix. i 3/1-1/16 5 5 A₁ = i = 1 • Find the real-valued solution to the initial value problem yí = ly/₂ = Use t as the independent variable in your answers. 31(t) = 11 y2(t) = -15 15 2 -5 -3 3y1 + 2y2, -5y1 - 3y2, y. and A₂ = -i 9 y₁ (0) = 11, Y2(0) = −15. 9 V2 = T 1 i 3/4 + 1/4 5 35The 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 -:.. ( poFind the eigenvalues and eigen functions of the Strum-Liouville problem u" + Au =0, 0sx3- Solve the problem P.D.E. u, = uxx 0Recommended textbooks for youCalculus: Early TranscendentalsCalculusISBN:9781285741550Author:James StewartPublisher:Cengage Learning
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