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# solving heat conduction equation watch

1. I have been asked to solve the following heat conduction equation:

over 0<x<3 and t>0 for the boundary conditions

u(0,t) = u(3,t) = 0

and the initial condition

u(x,0) = 5sin(4*pi*x)

Hint: The general solution of the ODE

is Bcos(lander.x) + Csin(lander.x) where lander, B and C are constants.

I have done the first few steps and have found that either Y(0) = Y(3) = 0 and G(t) does not equal 0.

I am stuck now, I am unsure how to progress further with this question. How do i find Y(x) and G(t)?
2. (Original post by sarah_vickers)
I have been asked to solve the following heat conduction equation:

over 0<x<3 and t>0 for the boundary conditions

u(0,t) = u(3,t) = 0

and the initial condition

u(x,0) = 5sin(4*pi*x)

Hint: The general solution of the ODE

is Bcos(lander.x) + Csin(lander.x) where lander, B and C are constants.

I have done the first few steps and have found that either Y(0) = Y(3) = 0 and G(t) does not equal 0.

I am stuck now, I am unsure how to progress further with this question. How do i find Y(x) and G(t)?
How do you normally solve these types of questions? I use separation of variable. Let

U(x,t)=T(t)X(x) and substitue into the heat equation to give 2X"(1/x)=T'(1/t) then let this to equal lamda^2 i.e. a constant and solve for the 2 equations,

2X"(1/x)=lamda^2 which you can use the hint to solve

T'(1/t)=lamda^2 which is a standard fode

sub in the contraints and you are done.

By the way what year if this material 1st or 2nd year?

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