It can be shown that the seperation constant must be negative or you will only get the trivial solution.

Here is the proof for homogenoious Dirichlet or Neumann boundry contdtions

after seperating we get

So we get

Now multiply this equation by and integrate the equation.

Now integrate the first term by parts with and to get

Now if we have the homogenious condtions mentioned above either

in either case the first term above is equal to zero when evaluated at the end points. So we get

Since both are always non negative the only when this integral can be zero is if

So the seperation constant must be negative in the heat equation.

This can be shown for the mixed and other boundry conditions but it is not as clean. I hope this helps.