Consider the second order linear ODE (1 - x²)y" - xy + X²y = 0 0 ≤ x ≤ 1/2, with boundary conditions y(0) = 0 y(1/2) = 0. (a) (b) (c) (d) form: Demonstrate that this ODE can be rewritten in Sturm-Liouville da (p(x) dy dx +q(x)y - X²r(x)y = 0, (미리)뿜) and identify p(x), q(x), and r(x). Characterize this boundary value problem as a regular or singular Sturm-Liouville problem. Solve for the eigenvalues and eigenfunctions (Hint: Consider the change of variable x = cos(0).) Suppose we consider this ODE on the interval -1 ≤ x ≤ 1. Again consider the change of variable suggested in part (c) and show that the ODE for this domain still has a discrete spectrum and compute the eigenvalues and eigenvectors. Note that for this case we are looking for solutions that are regular ±1 by which we mean the solution and all its derivatives are finite at at x = x = = ±1.
Consider the second order linear ODE (1 - x²)y" - xy + X²y = 0 0 ≤ x ≤ 1/2, with boundary conditions y(0) = 0 y(1/2) = 0. (a) (b) (c) (d) form: Demonstrate that this ODE can be rewritten in Sturm-Liouville da (p(x) dy dx +q(x)y - X²r(x)y = 0, (미리)뿜) and identify p(x), q(x), and r(x). Characterize this boundary value problem as a regular or singular Sturm-Liouville problem. Solve for the eigenvalues and eigenfunctions (Hint: Consider the change of variable x = cos(0).) Suppose we consider this ODE on the interval -1 ≤ x ≤ 1. Again consider the change of variable suggested in part (c) and show that the ODE for this domain still has a discrete spectrum and compute the eigenvalues and eigenvectors. Note that for this case we are looking for solutions that are regular ±1 by which we mean the solution and all its derivatives are finite at at x = x = = ±1.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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help me with part d please
Transcribed Image Text:Consider the second order linear ODE
(1 - x²)y" - xy + X²y = 0 0 ≤ x ≤ 1/2,
with boundary conditions
y(0) = 0 y(1/2) = 0.
(a)
(b)
(c)
(d)
form:
Demonstrate that this ODE can be rewritten in Sturm-Liouville
da (p(x) dy
dx
+q(x)y - X²r(x)y = 0,
(미리)뿜)
and identify p(x), q(x), and r(x).
Characterize this boundary value problem as a regular or singular
Sturm-Liouville problem.
Solve for the eigenvalues and eigenfunctions (Hint: Consider the
change of variable x = cos(0).)
Suppose we consider this ODE on the interval -1 ≤ x ≤ 1. Again
consider the change of variable suggested in part (c) and show that the ODE
for this domain still has a discrete spectrum and compute the eigenvalues and
eigenvectors. Note that for this case we are looking for solutions that are regular
±1 by which we mean the solution and all its derivatives are finite at
at x =
x = = ±1.
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