- #1
ZedCar
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Homework Statement
The one-dimensional heat diffusion equation is given by :
∂t(x,t)/∂t = α[∂^2T(x,t) / ∂x^2]
where α is positive.
Is the following a possible solution? Assume that the constants a and b can take any positive value.
T(x,t) = exp(at)cos(bx)
Homework Equations
The Attempt at a Solution
T(x,t) = exp(at)cos(bx)
∂T/∂t = a exp(at) cos(bx)
∂^2 T/∂x^2 = -b^2 exp(at) cos(bx)
As a, b and α are all positive this cannot be a solution.
A friend and I were working on this and got the answer above. Though as it was about a week ago I can't exactly remember what we've done here.
In the first step I think we've differentiated exp(at) with respect to t, treating the cos(bx) as a constant multiplier, which become a exp(at) cos(bx).
Is that all correct above?
Then we differentiated again with respect to t.
So wouldn't that become;
∂^2 T/∂x^2 = a^2 exp(at) cos(bx)
Why did we previously get ∂^2 T/∂x^2 = -b^2 exp(at) cos(bx)
Thanks!