Thread: Applying boundary conditions to heat equation

I follow the course for solving the heat equation. However, I cannot understand a possible case of semi-infinite.

The solution is

$$\phi(x) = c_1\cos (\sqrt{\lambda}x) + c_2 \sin(\sqrt{\lambda} x)$$


as we have to obtain $\lambda$ by applying the boundary conditions. All example are for $x=0$ and $x=L$. I am trying to figure out how to solve it for a simple set of boundary conditions:

$$u(x,t)=0, u(0,t)=10, u(\infty,t)=0$$

How do we apply $\phi(\infty) = 0$ to obtain $\lambda$?

Set the problem up for some fixed "L" and then take the limit as L goes to infinity.