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(Solved): Consider the following IBVP on the domain 0,\pi : x(x)T(t)x(x)5x_(n)(x)T_(n)(t)u_(tt)=64u_(\times ), ...
Consider the following IBVP on the domain 0,\pi :
x(x)T(t)x(x)5x_(n)(x)T_(n)(t)u_(tt)=64u_(\times ),00
u(0,t)=u_(x)(\pi ,t)=0,t>0
u(x,0)=sin((x)/(2))-3sin((5)/(2)x),0
(a) [10 points] Use the the method of separation of variables to derive a BVP in space satisfied
by x(x) and an ODE in time satisfied by T(t).
(b) [10 points] Solve the BVP in space found in (a) for x(x). Note that each viable separation
constant yields a linearly independent solution!
(c) [ 5 points] Solve the sequence of ODEs in time found in (a) associated with the x_(n)(x) solutions
in (b) for T_(n)(t).
(d) [5 points] Use the initial conditions to write down the solution to the IBVP.
Hint: the structure of the given initial data should make identifying coefficients a relatively
smooth experience. You may assume that two sine series expansions are equal provided that
their coefficients are equal.