The Question:
The temperature $u(x,t)$ in a semi-infinite conductor occupying $x \in [0,\infty)$ satisfies the equation
$$\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} \qquad x,t>0$$
The temperature is $0$ at $t=0$, i.e. $u(x,0)=0$.
For $t>0$, heat is supplied at $x=0$ at the constant flux
$$Q=-ku_x(0,t) \qquad k,Q \; \text{are constants}$$
By applying the Laplace Transform in the $t$ direction, find $u(x,t)$ at $x=0$ for $t>0$.
My Attempt:
I transformed the PDE, using the fact that $u(x,0)=0$:
$$p\hat u(x,p) = \frac{\partial ^2 \hat u}{\partial x^2}(x,p) \qquad p>0$$
and found the general solution:
$$\hat u(x,p) = A(p)e^{\sqrt p x}+B(p)e^{-\sqrt px}$$
Next, I transformed the boundary condition:
$$\frac{\partial \hat u}{\partial x}(0,p) = -\frac{Q}{k}p$$
BUT THE PROBLEM IS that there is only one boundary condition. Even if the question only asks for me to determine $u(x,t)$ at $x=0$, I still need one more boundary condition to determine both $A(p)$ and $B(p)$.
Am I misunderstanding something?