For the harmonic oscillator with mass
(a) compute the eigenvalues and associated eigenvectors;
(b) for each eigenvalue, pick an associated eigenvector
(c) for each solution derived in part (b), plot its solution curve in the y v-phase plane;
(d) for each solution derived in part (b), plot its
(e) for each solution derived in part (b), give a brief description of the behavior of the mass-spring system.
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Differential Equations
- Apply the eigenvalue method to find a general solution of the given system. For the given initial conditions, find also the corresponding particular solution. Use a computer system or graphing calculator to construct a direction field and typical solution curves for the given system. x'₁ = 4×₁ = 2x2, x' 2 = 16×1₁ − 4×2, ×₁ (0) = 5, x2 (0) = 4 What is the general solution in matrix form? x(t) =arrow_forwardConsider the Sturm-Liouville Problem?X²y" – 3xy'+ 4λy = 0 , (1) = 0 y(3) = 0 Find all eigenvalues and corresponding eigenfunctions.(b) Write the equation in Sturm-Liouville form, and determine the weight function.(c) Find the coefficient an, narrow_forwardFind the eigenvalues 1, and eigenfunctions y,(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.) y" + Ay = 0, y(0) = 0, y(n/3) = 0 1, = 4n?, Vn = sin(2nx), n= 1,2,3,.. a. A, = 36n?, V,=sin(6nx), n= 1,2,3,... A, = 25n?, V,= sin(5nx), n= 1,2,3,.. b. C. A, = 9n?, Vn = sin(3nx), n= 1,2,3,.. d.arrow_forward
- Find the eigenvalues and eigenfunctions of the boundary-value problemx2y′′+xy′+ 9λy= 0, y′(1) = 0,y(e) = 0. Put the equation in self-adjoint form, and give an orthogonality relation Show that if ym and yn are two eigenfunctions corresponding to two different eigenvalues λm and λn respectively, then ym and yn are linearly independentarrow_forwardUse the method of separation of variables to construct the energy eigenfunctions for the particle trapped in a 2D box. In other words, solve the equation: -h? ( 020, (x, y) a²¤n(x, y) En P, (x, y), 2m dx? dy? such that the solution is zero at the boundaries of a box of 'width' L, and 'height' Ly. You will see that the 'allowed' energies En are quantized just like the case of the 1D box. It is most convenient to to place the box in the first quadrant with one vertex at the origin.arrow_forwardConsider the linear system a. Find the eigenvalues and eigenvectors for the coefficient matrix. (-3+i) A₁ = 1 , vi b. Find the real-valued solution to the initial value problem [{ and A₂ = -3y1 - 2/2, 591 +33/2, Use t as the independent variable in your answers. vi(t) =(5-5/2i)e^(-it)(-3/5-1/5)+(5+5/2i)e^(it)(-3/5+i/5) 32(t)= (5-5/2)^(-it)+(5+5/2)^(it) 4 vo = 3/1 (0) = 6, 3/2 (0) = -5. (-3-1)/arrow_forward
- For the system of differential equations, = A₁, A₂ = U₁ = U2 = y' a) Find the characteristic polynomial of the matrix of coefficients A. CA(X) = [16 b) Find the eigenvalues of A. Enter the eigenvalues as a list in ascending order separated by commas. = -16 21 C1 c) Find the eigenvectors assuming u₁ is the eigenvector associated with the smaller eigenvalue X₁ and u2 is the eigenvector associated with the larger eigenvalue X₂ . Enter the eigenvectors as a matrix with an appropriate size. 14 19 y d) Determine a general solution to the system. Enter your answer in the format y(t) = c₁f₁(t)ví + c₂f₂(t)v₂ . y(t) = + C₂arrow_forward2) Find the eigenvalues and associated eigenfunctions for the equation Y" + AY = 0, λ>0 and Y(0)= 0 , Y(8) = 0.arrow_forward(VII). (12') Find the all eigenvalues and the corresponding eigenspace of matrix A, 0 0 where A = -1 1 3 Further factor the matrix A into a product XDX¯', where 1 1 D is diagonal.arrow_forward
- Find the eigenvalues 1, and eigenfunctions y,(x) for the given boundary-value problem. (Give your answers in terms of n, making sure that each value ofn corresponds to a unique eigenvalue.) x²y" + xy' + Ay = 0, y'(e-l) = 0, y(1) = 0 n = 1, 2, 3, ... Yn(x) = n = 1, 2, 3, ... Need Help? Read It Watch It search 0 耳 回 DII PrtScn Home End F2 F3 F4 F5 F6 F7 F8 F9 2$ & 3 4 7 8. 9. R Yarrow_forwardFind the eigenvalues 1, and eigenfunctions y,(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.) x?y" + xy' + ly = 0, y'(e-1) = 0, y(1) = 0 n = 1, 2, 3, ... Yo(x) n = 1, 2, 3,...arrow_forwardFor the system of differential equations, X1, X2 a) Find the characteristic polynomial of the matrix of coefficients A. CA(X) b) Find the eigenvalues of A. Enter the eigenvalues as a list in ascending order separated by commas. U₁ = U2 = = y' = c) Find the eigenvectors assuming u₁ is the eigenvector associated with the smaller eigenvalue X₁ and u2 is the eigenvector associated with the larger eigenvalue A2. 5 2 - 1 Y2 (t) d) Determine two linearly independent solutions to the system. Enter the first solution in the format y₁ (t) y = = y₁ (t) Enter the second solution in the format y₂(t) = ƒ2(t) (v3h₁(t) + v₁h₂(t)) = : fi(t) (vigi(t) – v2g2(t)) . +arrow_forward
- Calculus For The Life SciencesCalculusISBN:9780321964038Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.Publisher:Pearson Addison Wesley,