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Unformatted text preview: . This is an
example of a boundary value problem (BVP) for a fourth-order linear ODE.
This is a boundary value problem because:
Observe that y(t) = 0 is a solution of this problem for all values of and l. A
more interesting problem is to determine y and those values of and l for which
non-trivial (y(t) 6= 0) solutions exist. As such, this BVP is also a di erential
eigenvalue problem. Nontrivial solutions y(t) are called eigen functions and their
corresponding values of are eigenvalues. y(t)
t Figure 1.1.2: Buckling of an elastic column.
4. Pendelum Oscillations ( 1], Section 1.3). The position (x(t) y(t)) at time t of a
particle of mass m oscillating on a pendelum of length l (Figure 1.1.3) is
m d x = ;T sin = ; T x
l m d y = mg ; T cos = mg ; T y
2 t>0 where g is the acceleration of gravity, T (t) is the tension in the string, and (t) is
the angle of the pendelum relative to the vertical at time t (Figure 1.1.3). These
equations are, however, insu cient to guarantee that the particle stays on the
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This document was uploaded on 03/16/2014 for the course CSCI 6820 at Rensselaer Polytechnic Institute.
- Spring '14