Consider the fourth-order boundary-value problem p'' (x) – a*@(x) = 0 p(0) = p'(0) = 0 P(1) = p"(1) = 0 where a > 0. Show that there is a non-trivial solution p(x) if and only if tan a = tanh a. Note: This boundary-value problem arises in modeling the vibrations of a beam that is clamped at one end and simply supported at the other. Hint: Write the general solution of o' (x) – a*p(x) = 0 as p(x) = C cosh ax + C2 sinh ax + c3 cos ax + C4 sin ax 2.
Consider the fourth-order boundary-value problem p'' (x) – a*@(x) = 0 p(0) = p'(0) = 0 P(1) = p"(1) = 0 where a > 0. Show that there is a non-trivial solution p(x) if and only if tan a = tanh a. Note: This boundary-value problem arises in modeling the vibrations of a beam that is clamped at one end and simply supported at the other. Hint: Write the general solution of o' (x) – a*p(x) = 0 as p(x) = C cosh ax + C2 sinh ax + c3 cos ax + C4 sin ax 2.
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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Transcribed Image Text:2.
Consider the fourth-order boundary-value problem
p"(x) – a*p(x) = 0
P(0) = p'(0) = 0
φ(1) φ"(1) 0
where a > 0. Show that there is a non-trivial solution p (x) if and only if tan a = tanh a.
Note: This boundary-value problem arises in modeling the vibrations of a beam that is clamped at one
end and simply supported at the other.
Hint: Write the general solution of p' (x) – a*q(x) = 0 as
p (x) = c1 cosh ax + C2 sinh ax + c3 cos ax + C4 sin ax
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