PDE, heat equation lambda =,<,> 0 question

[kevin_tee]

So I have been studying solving separation of variable, heat equation and came across 2 set of lambda equation.

and

lambda = 0 have the same equation. Is it different?

 

[Ssnow]

Hint, answer to this question:what is the simplest function such that $\frac{d^2}{dx^2}X(x)=0$ ?