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3: (a) Solve the heat equation

`(delu)/(delt)=(del^(2)u)/(delx^(2))`

for

`u=u(x,t)`

with

`0<=x<=\pi `

and

`t>=0`

satisfying the insulated ends condition

`(delu)/(delx)(0,t)=(delu)/(delx)(\pi ,t)=0`

for all

`t>=0`

and the initial condition

`u(x,0)=x^(2)`

for all

`0<=x<=\pi `

. (b) Give a fairly accurately sketch of the graphs of

`u=u(x,t)`

(in the

`xu`

-plane) for

`t=0,(1)/(2),1,10`

.