Question

show that u(x,t)-sin(nx)e^(-n^(2)t) satisfies the heat equation df/dt = d^2f/dx^2 for any constant n?

ok so what i think i have to do is find the partial and the second partial derivatives of the equation...then what do i do?
show that u(x,t)-sin(nx)e^(-n^(2)t) satisfies the heat equation df/dt = d^2f/dx^2 for any constant n?

ok so what i think i have to do is find the partial and the second partial derivatives of the equation...then what do i do?
@Mathematics

Answer 1

Then see if they are equal!

Answer 2

so after solving both du/dt and d^2u/dx^2 i got them both to be -n^2sin(nx)e^(-n^(2)t)...
is that what i am supposed to get?

Answer 3

You have \(u(x,t)=\sin(nx)e^{-n^2t} \implies \frac{\partial u}{\partial t}=\sin(nx)(-n^2e^{-n^2t})=-n^2\sin(nx)e^{-n^2t}\). Now we should find the second partial derivative with respect to \(x\) and see if they are the same. \(\frac{\partial u}{\partial x}=n\cos(nx)e^{-n^2t} \implies \frac{\partial^2 u}{\partial x^2}=-n^2 \sin(nx)e^{-n^2t}.\) So, \(\frac{\partial^2 u}{\partial x^2}=\frac{\partial u}{\partial t}\) and therefore it satisfies the heat equation.

Answer 4

Yeah!

Answer 5

cool thanks AnwarA! :)

Answer 6

You're welcome!