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Differential equations

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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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 ...

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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