Finding the solution to a differential equation

Question

Note: $u$ is independent of $t$.

Let $u_l$ be the solution for $u_t = 2u_{xx}$ where $u(0)=0$. Let $u_r$ be the solution for $u_t = u_{xx}$ where $u(5)=10$.

However, $u_l(3)=u_r(3)$ and $2\partial_x u_l(3) = \partial_x u_r (3)$.

Answer 1

$u_l$ and $u_r$ are not unique: $$\begin{align} u_l(x)&=A\,x\\ u_r(x)&=10+B(x-5). \end{align} $$ When $A=B=2$ both coincide.