Heat Equation, possible solutions 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?

*

*Plug $u(x,t)=T(t)\,X(x)$ into the heat equation.

*Obtain an equation where on the left hand side you have $T$ and $T'$ and on the right hand side $X$ and $X'$.

*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?

*Obtain a second order ordinary differential linear equation for $X$ and solve using the boundary conditions.

*Obtain a first order ordinary differential linear equation for for $T$ and solve.

