Differential equations
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Consider the following problem for u = u(x,t):
,
,
a) Seeking a solution of this problem of the form
,
show that f and g satisfy the coupled system
; where and ;
, ; , .
b) Eliminating g between the differential equations in a), show that f satisfies
c) Assuming that
and noting that
conclude that
d) Applying the boundary conditions for f in a), deduce that
where
e) Now use a) to show that
.
f) Applying the boundary conditions for g in a), finally conclude that
and hence obtain the result.
since
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Solution Summary
This shows how to solve a differential equation problem using a series of proofs. The boundary conditions applied are determined.
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Consider the following problem for u = u(x,t):
,
,
a) Seeking a solution of this problem of the form
,
show that f and g satisfy the coupled system
; where and ;
, ; , .
Now,
Dividing throughout by
=
Comparing the cos terms, and comparing the sin terms,
For u(0, t) = cos(t), f(0) = 1 and g(0) = 0
b) Eliminating g between the differential equations in a), show ...
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