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

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

zOjJ6G1.png

and
GqgLWND.png

lambda = 0 have the same equation. Is it different?
 
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Hint, answer to this question:what is the simplest function such that ##\frac{d^2}{dx^2}X(x)=0## ?
 

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