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2: (a) Solve the wave equation

`(del^(2)u)/(delt^(2))=(1)/(4)(del^(2)u)/(delx^(2))`

for

`u=u(x,t)`

with

`0<=x<=2`

and

`tinR`

satisfying the fixed ends condition

`u(0,t)=u(2,t)=0`

for all

`tinR`

and the initial conditions

`u(x,0)=(sin\pi x)(1+cos\pi x)`

and

`(delu)/(delt)(x,0)=0`

for all

`0<=x<=2`

. (b) Find a constant

`c`

and function

`g(x)`

such that

`u(x,t)=g(x+ct)+g(x-ct)`

for all

`x,t`

. (c) By plotting points, accurately sketch the graphs

`u=u(x,t)`

(in the

`xu`

-plane) for

`t=0,(1)/(2),1,(3)/(2),2`

.