the ends of a string of length L are secured at both ends. Its initial shape is straight but it is vibrating with initial velocity f(x)=x(L-x).choose the PDE and boundary/initial conditions that model this scenario. Find Eigenfunctions for ).The problem splits into cases based on the sign of A(Notation: For the cases below, use constants a and b)• Case 1: A = 0X(x)Plugging the boundary values into this formula gives0 = X(0)%3D0 = X(6)So X(x) =which means u(x, t) =We can ignore this case.• Case 2: A =-y² <0(In your answers below use gamma instead of lambda)X(x) =Plugging the boundary values into this formula gives0 – X(0)0 = X(6) =So X(x) =which means u(x, t) =We can ingore this case.• Case 3: A= >X(x) =(In your answers below use gamma instead of lambda)Plugging in the boundary values into this formula gives0 = X(0) =%3D0 = X(6)Which leads us to the eigenvalues A, = Yn where Yn =and eigenfunctions X,(z) =(Notation: Eigenfunctions should not include any constants a or b.)

Let the given equation be, x''-λx=0.

The problem splits into cases based on the sign of ). (Notation: For the cases below, use constants a and b) • Case 1: A = 0 X(z) = %3D Plugging the boundary values into this formula gives 0 X(0) = %3D %3D 0 = X(6) = So X(x) = %3D which means u(r, t) = %3D We can ignore this case. Case 2: A = -y? < 0 (In your answers below use gamma instead of lambda) X(x) = %3D Plugging the boundary values into this formula gives 0 = X(0) = 0 = X(6) So X(x) = %3D %3D which means u(x, t) = We can ingore this case. Case 3: A = y >0 (In your answers below use gamma instead of lambda) X(x) = %3D Plugging in the boundary values into this formula gives 0 = X(0) = 0 = X(6) =

The problem splits into cases based on the sign of ). (Notation: For the cases below, use constants a and b) • Case 1: A = 0 X(z) = %3D Plugging the boundary values into this formula gives 0 X(0) = %3D %3D 0 = X(6) = So X(x) = %3D which means u(r, t) = %3D We can ignore this case. Case 2: A = -y? < 0 (In your answers below use gamma instead of lambda) X(x) = %3D Plugging the boundary values into this formula gives 0 = X(0) = 0 = X(6) So X(x) = %3D %3D which means u(x, t) = We can ingore this case. Case 3: A = y >0 (In your answers below use gamma instead of lambda) X(x) = %3D Plugging in the boundary values into this formula gives 0 = X(0) = 0 = X(6) =

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter7: Eigenvalues And Eigenvectors

Section7.1: Eigenvalues And Eigenvectors

Problem 65E

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