Heat Equation, possible solutions

Question

NOTE: This is a homework problem. Please do not solve.

I was given a problem that asked me to find a function of the form $u_n(x,t)=\chi_n(x) \cdot T_n(t) $ that solves the heat equation with the following conditions:

$u_t = u_{xx}\\ u(0,t)=0\\ u_x(1,t)=0$

That is all of the information, but I am unsure how to solve such a problem without an initial heat distribution.

I considered using the final condition, but all that tells me is that $u(1,t) = f(t)$ for some function $f(t)$, not what that $f(t)$ could be.

Is a solution even possible based on this information?

Answer 1

1. Plug $u(x,t)=T(t)\,X(x)$ into the heat equation.
2. Obtain an equation where on the left hand side you have $T$ and $T'$ and on the right hand side $X$ and $X'$.
3. If a function that depends only on $t$ is equal to a function that depends only on $x$, what type of functions can they be?
4. Obtain a second order ordinary differential linear equation for $X$ and solve using the boundary conditions.
5. Obtain a first order ordinary differential linear equation for for $T$ and solve.