Advanced Engineering Mathematics
6th Edition
ISBN: 9781284105902
Author: Dennis G. Zill
Publisher: Jones & Bartlett Learning
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Chapter 3.9, Problem 15E
To determine
The eigenvalues and eigenfunctions for the given boundary-value problem.
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1. Find the eigenvalues and the eigenfunctions of the following problems:
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(ii) x²y"(x) + xy'(x) + Ay(x) = 0; '(1) = 0, y'(e) = 0.
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Q.6 Use eigenfunction expantion methodto solve:
-(xy')-
y(1) = e, y(e)=0 :
.........
2. Find the eigenvalues and the corresponding eigenfunctions of the following boundary
value problems. (Use real coefficients.)
(a) y"+ Ay = 0 with y(0) = 0 and y'(T) = 0.
(b) y" + Ay = 0 with y'(0) =0 and y(L) = 0.
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Chapter 3 Solutions
Advanced Engineering Mathematics
Ch. 3.1 - Prob. 1ECh. 3.1 - Prob. 2ECh. 3.1 - Prob. 3ECh. 3.1 - Prob. 4ECh. 3.1 - Prob. 5ECh. 3.1 - Prob. 6ECh. 3.1 - Prob. 7ECh. 3.1 - Prob. 8ECh. 3.1 - Prob. 9ECh. 3.1 - Prob. 10E
Ch. 3.1 - Prob. 11ECh. 3.1 - Prob. 12ECh. 3.1 - Prob. 13ECh. 3.1 - Prob. 14ECh. 3.1 - Prob. 15ECh. 3.1 - Prob. 16ECh. 3.1 - Prob. 17ECh. 3.1 - Prob. 18ECh. 3.1 - Prob. 19ECh. 3.1 - Prob. 20ECh. 3.1 - Prob. 21ECh. 3.1 - Prob. 22ECh. 3.1 - Prob. 23ECh. 3.1 - Prob. 24ECh. 3.1 - Prob. 25ECh. 3.1 - Prob. 26ECh. 3.1 - Prob. 27ECh. 3.1 - Prob. 28ECh. 3.1 - Prob. 29ECh. 3.1 - Prob. 30ECh. 3.1 - Prob. 31ECh. 3.1 - Prob. 32ECh. 3.1 - Prob. 33ECh. 3.1 - Prob. 34ECh. 3.1 - Prob. 35ECh. 3.1 - Prob. 36ECh. 3.1 - Prob. 37ECh. 3.1 - Prob. 38ECh. 3.1 - Prob. 40ECh. 3.1 - Prob. 42ECh. 3.2 - Prob. 1ECh. 3.2 - Prob. 2ECh. 3.2 - Prob. 3ECh. 3.2 - Prob. 4ECh. 3.2 - Prob. 5ECh. 3.2 - Prob. 6ECh. 3.2 - Prob. 7ECh. 3.2 - Prob. 8ECh. 3.2 - Prob. 9ECh. 3.2 - Prob. 10ECh. 3.2 - Prob. 11ECh. 3.2 - Prob. 12ECh. 3.2 - Prob. 13ECh. 3.2 - Prob. 14ECh. 3.2 - Prob. 15ECh. 3.2 - Prob. 16ECh. 3.2 - Prob. 17ECh. 3.2 - Prob. 18ECh. 3.2 - Prob. 19ECh. 3.2 - Prob. 20ECh. 3.2 - Prob. 21ECh. 3.2 - Prob. 22ECh. 3.2 - Prob. 23ECh. 3.2 - 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Prob. 45ECh. 3.4 - Prob. 46ECh. 3.4 - Prob. 47ECh. 3.4 - Prob. 48ECh. 3.5 - Prob. 1ECh. 3.5 - Prob. 2ECh. 3.5 - Prob. 3ECh. 3.5 - Prob. 4ECh. 3.5 - Prob. 5ECh. 3.5 - Prob. 6ECh. 3.5 - Prob. 7ECh. 3.5 - Prob. 8ECh. 3.5 - Prob. 9ECh. 3.5 - Prob. 10ECh. 3.5 - Prob. 11ECh. 3.5 - Prob. 12ECh. 3.5 - Prob. 13ECh. 3.5 - Prob. 14ECh. 3.5 - Prob. 15ECh. 3.5 - Prob. 16ECh. 3.5 - Prob. 17ECh. 3.5 - Prob. 18ECh. 3.5 - Prob. 19ECh. 3.5 - Prob. 20ECh. 3.5 - Prob. 21ECh. 3.5 - Prob. 22ECh. 3.5 - Prob. 23ECh. 3.5 - Prob. 24ECh. 3.5 - Prob. 25ECh. 3.5 - Prob. 26ECh. 3.5 - Prob. 27ECh. 3.5 - Prob. 28ECh. 3.5 - Prob. 29ECh. 3.5 - Prob. 30ECh. 3.5 - Prob. 31ECh. 3.5 - Prob. 32ECh. 3.5 - Prob. 33ECh. 3.5 - Prob. 34ECh. 3.6 - Prob. 1ECh. 3.6 - Prob. 2ECh. 3.6 - Prob. 3ECh. 3.6 - Prob. 4ECh. 3.6 - Prob. 5ECh. 3.6 - Prob. 6ECh. 3.6 - Prob. 7ECh. 3.6 - Prob. 8ECh. 3.6 - Prob. 9ECh. 3.6 - Prob. 10ECh. 3.6 - Prob. 11ECh. 3.6 - Prob. 12ECh. 3.6 - Prob. 13ECh. 3.6 - Prob. 14ECh. 3.6 - Prob. 15ECh. 3.6 - Prob. 16ECh. 3.6 - 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Prob. 8ECh. 3.10 - Prob. 9ECh. 3.10 - Prob. 10ECh. 3.10 - Prob. 11ECh. 3.10 - Prob. 12ECh. 3.10 - Prob. 13ECh. 3.10 - Prob. 14ECh. 3.10 - Prob. 15ECh. 3.10 - Prob. 16ECh. 3.10 - Prob. 17ECh. 3.10 - Prob. 18ECh. 3.10 - Prob. 19ECh. 3.10 - Prob. 20ECh. 3.10 - Prob. 21ECh. 3.10 - Prob. 22ECh. 3.10 - Prob. 23ECh. 3.10 - Prob. 24ECh. 3.10 - Prob. 25ECh. 3.10 - Prob. 26ECh. 3.10 - Prob. 27ECh. 3.10 - Prob. 28ECh. 3.10 - Prob. 29ECh. 3.10 - Prob. 30ECh. 3.10 - Prob. 31ECh. 3.10 - Prob. 32ECh. 3.10 - Prob. 33ECh. 3.10 - Prob. 34ECh. 3.10 - Prob. 35ECh. 3.10 - Prob. 36ECh. 3.10 - Prob. 37ECh. 3.10 - Prob. 38ECh. 3.10 - Prob. 39ECh. 3.10 - Prob. 40ECh. 3.10 - Prob. 41ECh. 3.10 - Prob. 42ECh. 3.10 - Prob. 43ECh. 3.10 - Prob. 44ECh. 3.10 - Prob. 45ECh. 3.10 - Prob. 46ECh. 3.11 - Prob. 7ECh. 3.11 - Prob. 14ECh. 3.11 - Prob. 15ECh. 3.11 - Prob. 16ECh. 3.11 - Prob. 20ECh. 3.11 - Prob. 21ECh. 3.12 - Prob. 1ECh. 3.12 - Prob. 2ECh. 3.12 - Prob. 3ECh. 3.12 - Prob. 4ECh. 3.12 - Prob. 5ECh. 3.12 - Prob. 6ECh. 3.12 - 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- 1. Find the eigenvalues and eigenfunctions of the boundary value problem y" + y' + Xy = 0, y(0) = 0, y(2) = 0 The solution requires analysis of different cases of ): specifically, for which value(s) of A are non-trivial solutions obtained?arrow_forward2. Find the eigenvalues and eigenfunctions of the boundary value problem. y" - Xy = 0, y (0) = 0, y'(L) = 0. (Note the second value is a derivative!)arrow_forward1. Find the eigenvalues and the eigenfunctions of the following problems: (i) y"(x)+ Ay(x) = 0; y(0) = y(1), y'(0) = y'(1). (ii) x²y"(x)+ xy'(x) + Ay(x) = 0; y (1) = 0, y'(e) = 0.arrow_forward
- 3) Show that the large eigenfunctions of the problem y″ + X(x + π)¹y = 0, y(0) = y(t) = 0, are given approximately by λ 9n² 4974 for large integers n and find the corresponding eigenfunctions.arrow_forwardSuppose that the differential equation x' = Ax is such that A is a 2 × 2 matrix with (3) eigenvalues A1 = 5 and A2 = 3 and corresponding eigenvectors vi = and v2 = the solution that satisfies the initial condition x(0) = () Explain what happens to the trajectory of this solution as t → 0. Circle one of the items below. (6) i. becomes parallel to the vector vị = ii. becomes parallel to the vector v2 = iii. tends towards the point (0,0). iv. Spirals away from the origin v. none of the abovearrow_forward3. (a) Find the eigenvalues and eigenfunctions of the boundary-value problem x2y'' + xy' + (lamda)y = 0, y(1) = 0, y(5) = 0 (b) Put the differential equation in self-adjoint form. (c) Give an orthogonality relation.arrow_forward
- The boundary value problem y" + λy = 0 y'(0) = 0 y' (L) = 0 has a solution : = 0 y(x) = 1. Find all other eigenvalues and eigenfunctions, λn Yn (x) for positive integers n.arrow_forwardFind the solution of the initial value problem Y' = AY,Y (0) = [1, 2] if we know that A has eigenpairs (2, [1, –1]) and (2, [0, 1])arrow_forward4. Suppose the characteristic equation of a second-order Cauchy-Euler equation has a repeated root r = c E R. (a) What's one solution of this ODE? Call this y₁. (b) To find a second solution that is linearly independent from the first, suppose y2 = u - Y1, where u = u(x). Solve for u(x) and then find y2(x). (c) Show that y₁ and y2 are indeed linearly independent.arrow_forward
- 13 14. . In Problems 15-26, solve by the method of eigenfunction expansions yte-d ve obtain 15. y y(0) =0, y(1)-0arrow_forward9. Form the differential equation of the three-parameter family of conics y = ae* + be2x + ce¬3x where a, b and c are arbitrary constants.arrow_forwardplease answer 2 and 5arrow_forward
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