Given: Heat equation: $\frac{\partial u}{\partial \tau }=\frac{\partial^2 u}{\partial x^2}$.
Initial condition: $u(x,0)=u_{0}(x)$
Appropriate boundary conditions.
We assume that $u_{1}(x,\tau)$ and $u_{2}(x,\tau)$ are solutions.
Considering that $u(x,\tau)=u_{1}(x,\tau)-u_{2}(x,\tau)$, how do we check that $u$ satisfies the heat equation? What is the initial value?
Check the putative solution and see if it satisfies the heat equation: $$ \frac{\partial }{\partial \tau}(u_1(x,\tau)-u_2(x,\tau))=\frac{\partial u_1}{\partial \tau}-\frac{\partial u_2}{\partial \tau} $$ and $$ \frac{\partial^2 }{\partial x^2}(u_1(x,\tau)-u_2(x,\tau))=\frac{\partial^2 u_1}{\partial x^2}-\frac{\partial^2 u_2}{\partial x^2} $$ but by assumption that $u_1$ and $u_2$ are solutions, we have $$ \frac{\partial u_1}{\partial \tau}=\frac{\partial^2 u_1}{\partial x^2}\\ \frac{\partial u_2}{\partial \tau}=\frac{\partial^2 u_2}{\partial x^2} $$ and thus $$ \frac{\partial u_1}{\partial \tau}-\frac{\partial u_2}{\partial \tau}= \frac{\partial^2 u_1}{\partial x^2}-\frac{\partial^2 u_2}{\partial x^2} $$ as required.
Finally, for the boundary conditions, note that $$ u_1(x,0)-u_2(x,0)=u_0(x)-u_0(x)=0 $$