8. For which values of does the boundary-value problem y" – 2y'+(1+1)y=0; y(0)=0, y(1)=0 have a...
(1 point) Determine the values of a (eigenvalues) for which the boundary-value problem y + y = 0, 0 < x < 8 y(0) = 0, y'(8) = 0 has a non-trivial solution. = an ((2n-1)^2pi^2)/256 ,n= 1, 2, 3, ... Your formula should give the eigenvalues in increasing order. The eigenfunctions to the eigenvalue an are Yn = Cn* sin ((2n-1) pi n/16) where Cn is an arbitrary cons
Problem 11. 12 marks] Consider the following two-point boundary value problem: y" + y' + ßy = 0, y(0) = 0, y(1) = 0, where ß is a real nurnber. we know the problern has a trivial solution, i.e. y(x) = 0, Discuss how the value of B influences the nontrivial solutions of the boundary value problem, and get the nontrivial solutions (Find all the real eigenvalues β and the corresponding eigenfunctions.)
Problem 11. 12 marks] Consider the following two-point...
(1 point) Determine the values of (eigenvalues) for which the boundary-value problem g” + y = 0, 0 < x < 4 y(0) = 0, y' (4) = 0 has a non-trivial solution. An = a , n=1,2,3,... Your formula should give the eigenvalues in increasing order. The eigenfunctions to the eigenvalue in are Yn = Cn* sin(n*pi/2*x) where On is an arbitrary constant.
Find the eigenvalues and eigenfunctions for the following
boundary-value problem.
xạy"+xy'+2y = 0, y'le')=0, y(1) =0)
differential equations
.. Boundary Value. Solve the following: y" + 2y' - 5y = 0, y(0) = 0, y'(1) = 0 F. Boundary Value. Solve the following: y" + 2y' - 3y = 9x, y(0) = 1, y'(1) = 2
Problem #8: Find the eigenfunctions for the following boundary value problem. x2y"-19xy(100 A)y = 0. y(e) = 0, y(1) = 0. In the eigenfunction take the arbitrary constant (either c1 or c) from the general solution to be 1 Enter your answer as a symbolic function of x.n, as in these examples Problem #8: Do not include 'yin your answer.
Problem #8: Find the eigenfunctions for the following boundary value problem. x2y"-19xy(100 A)y = 0. y(e) = 0, y(1) =...
Consider the following boundary value problem, y" +(+5) y = 0, y'() = 0, y(9) = 0 (a) Find the eigenvalues. (b) Find the eigenfunctions. Take the arbitrary constant (either cu or c) from the general solution to be 1. Consider the following boundary value problem, y" + (8 + 5) y = 0, y'(o) = 0, 9) = 0 (a) Find the eigenvalues. (b) Find the eigenfunctions. Take the arbitrary constant (either cy or c2) from the general solution...
Problem 2. (a) Solve the initial value problem I y' + 2y = g(t), 1 y(0) = 0, where where | 1 if t < 1, g(t) = { 10 if t > 1 (t) = { for all t. Is this solution unique for all time? Is it unique for any time? Does this contradict the existence and uniqueness theorem? Explain. (b) If the initial condition y(0) = 0 were replaced with y(1) = 0, would there necessarily be...
2. The boundary value problem y" + ly = 0, y'(0)= 0 , y'(1) = 0) has normalized eigenfunctions 6(x)=1, 0,,(x) = V2 cos nix, n=1,2,3,... a. Using the method of eigenfunction expansion, solve the boundary value problem y " + 8y = x , y'(0)= 0, y'(1)=0 Set up, but do not evaluate, the required integrals. b. Determine how many solutions the below boundary value problem has. y" + 257² y = sinº 5ax , y(0) = 0 ,...
Find the solution of the given initial value problem. y" + 2y' +2y = cost+8(-5); y(0) = 3, y'(0) = 5 ° 20) = us20e" sin + + cost ( +ş) + sint (36+}) x() ==««n6e8cose + cost (3e* +) + sint (80* + }) 20 = usz beé" sin + sing (54* +5.) +cos (34++}) ° 40 = =uaz(Dei* cost + cost ("* + 5 ) + sint (3*+ }) 209 = 192(e“ cose + cost (* +) +sint(****+})