a: . b: 5) For the Sturm-Liouville Eigenvalue Problem d dx 11 [p(x)] do +q(x)¢ + \o(x)¢ = 0, dx ap(a) + Bo' (a) = 0, yo(b) + 80' (b) = 0, with p(x) > 0 and σ(x) > 0 over a 0 and q(x) ≤0 are satisfied by all eigenfunctions over a
a: . b: 5) For the Sturm-Liouville Eigenvalue Problem d dx 11 [p(x)] do +q(x)¢ + \o(x)¢ = 0, dx ap(a) + Bo' (a) = 0, yo(b) + 80' (b) = 0, with p(x) > 0 and σ(x) > 0 over a 0 and q(x) ≤0 are satisfied by all eigenfunctions over a
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![a: .
b:
5)
For the Sturm-Liouville Eigenvalue Problem
d
dx
11 [p(x)]
do
+q(x)¢ + \o(x)¢ = 0,
dx
ap(a) + Bo' (a) = 0,
yo(b) + 80' (b) = 0,
with p(x) > 0 and σ(x) > 0 over a <x<b, the Rayleigh Quotient can be used
to estimate the size of the smallest eigenvalue:
A₁ = min
-pool + P(')² - qo² dx
So o²σ dx
Here, the minimum is taken over all possible functions that satisfy the boundary
conditions. We recover the smallest eigenvalue exactly if we insert the corre-
sponding eigenfunction 1.
Show that if the conditions
-po
do
dx a
>0 and q(x) ≤0
are satisfied by all eigenfunctions over a <x<b, then all eigenvalues are
guaranteed to be non-negative.
Determine a reasonably accurate upper bound on the lowest eigenvalue
for
o" - x²+16= 0,
o'(0) = 0 = (1).
HINT: Remember that the n-th eigenfunction has n-1 zeroes in the range
a < x < b. So for A1, try a trial function that matches the given boundary
conditions but does not have any zeros in the range 0 < x < 1.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb9ca0fe0-8326-4567-9bb6-5d60d0a741b4%2F913e0a57-c375-407d-9332-45c780d22bec%2Fwzdz9u_processed.jpeg&w=3840&q=75)
Transcribed Image Text:a: .
b:
5)
For the Sturm-Liouville Eigenvalue Problem
d
dx
11 [p(x)]
do
+q(x)¢ + \o(x)¢ = 0,
dx
ap(a) + Bo' (a) = 0,
yo(b) + 80' (b) = 0,
with p(x) > 0 and σ(x) > 0 over a <x<b, the Rayleigh Quotient can be used
to estimate the size of the smallest eigenvalue:
A₁ = min
-pool + P(')² - qo² dx
So o²σ dx
Here, the minimum is taken over all possible functions that satisfy the boundary
conditions. We recover the smallest eigenvalue exactly if we insert the corre-
sponding eigenfunction 1.
Show that if the conditions
-po
do
dx a
>0 and q(x) ≤0
are satisfied by all eigenfunctions over a <x<b, then all eigenvalues are
guaranteed to be non-negative.
Determine a reasonably accurate upper bound on the lowest eigenvalue
for
o" - x²+16= 0,
o'(0) = 0 = (1).
HINT: Remember that the n-th eigenfunction has n-1 zeroes in the range
a < x < b. So for A1, try a trial function that matches the given boundary
conditions but does not have any zeros in the range 0 < x < 1.
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