Do only first question... Part 1: Find the Eigenfunction ExpansionConsider the function f defined on the interval [0, 6] as follows,23x = [0,3],2,x = [3, 6].Find the coefficients Cn of the eigenfunction expansion of function f,f(x)=X,∞f(x) = Σ Cn Yn(x),n=1Cn =where yn, for n = 1, 2, 3, . are the unit eigenfunctions of the Regular Sturm-Liouville system-y" = λy,y(0) = 0,Note: Label your eigenfunctionsy(6) = 0.so the eigenfunction for the lowest eigenvalue corresponds to n = 1.M

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Answered: ▾ Part 1: Find the Eigenfunction…

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter4: Eigenvalues And Eigenvectors

Section4.6: Applications And The Perron-frobenius Theorem

Problem 70EQ

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eigenvalues 4 -1 - 2 To To 1 -2 - 1 8 3 CỤ OTC 19 0 2 3 -2 -3 1-11 5 -2 0 4 -2 421O 1 -2 4 0
Consider the initial value problem for the vector-valued function x, -1 x' = Ax, A = 1 x(0) = Find the eigenvalues A1, A2 and their corresponding eigenvectors v1, V2 of the coefficient matrix A. (a) Eigenvalues: (if repeated, enter it twice separated by commas) d1, A2 = 1, 1 Σ (b) Eigenvector for A1 you entered above: Vi = Σ (c) Either the eigenvector for X2 you entered above or the vector w computed with vị entered above in case of repeated eigenvalue: Either v2 or w = Σ (d) Find the solution x of the initial value problem above. x(t) = Σ Note: To enter the vector (u, v) type .
Explain this step just
For 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)) . +
Using the attached theorms, Solve dx/dt =10x-20y , dy/dt=8x-18y
Find the Laplace transform of the solution of initial value problem Select one: O A. None O B. O C. O D. O E. 6s +3 (8² - 9) (s - 1) 3 8² - 9 1 4s +3 3 (s - 1)(s + 3) g2 y" + 2y' - 3y = 6e³t, y(0) = 0, y'(0) = 1
Let z1 = -2 + 1li, z2 = 2 – i. Showing the details of your work, find, in the form x + iy: Re (z3). (Re z1) 7122, (2122) Re (1/23). 1/Re (z3 (z1 - 22/16, (z1/4 – 22/4)2 Z1/22. z2/21 (21 + z2)(21 – 32). zỉ – 3 Z1/22, (71/22) 4 (Z1 + 22)/(71 – 2)
X1(t) x2(t) a Suppose x'(t) = Ax(t) where A = ), a, b, c, d e R and æ(t) Use the d definition of linear independence to justify each of the following. Recall that if v E R" = 0, then each component of v must equal 0. (a) Suppose the eigenvalues of A are distinct and real-valued. Show that the terms in the solution x(t) are linearly independent. (b) Suppose there is only one eigenvalue of A. Show that the terms in the solution x(t) are linearly independent. (Hint: There are two casees here linearly independent eigenvectors associated with that eigenvalue and the second case is when we cannot find two linearly independent eigenvalues) - one case when there are two (c) Suppose the eigenvalues of A are complex conjugates. Show that the terms in the solution x(t) are linearly independent.

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