Tiny question on differential (?) equations.

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The discussion revolves around the potential V(x) in an electric field, specifically at the point x=0 where the first derivative dV/dx equals zero, indicating a minimum. The second derivative d²V/dx² is given as kQ/(2√2 L³), suggesting that V(x) can be expressed in a quadratic form. It is questioned whether the constant @ should be kQ/(4√2 L³) instead of kQ/(√2 L³), prompting a consideration of the function's symmetry. The user also raises the issue of whether V(x) is an odd or even function by examining V(-x). The conversation highlights the relationship between the potential's derivatives and its functional form in the context of differential equations.
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For the potential in an electric field is given:

http://img402.imageshack.us/img402/4162/naamloos8zn.gif

At x=0 V(x) is at a minimum so:

dV/dx =0 and d2V/dx2= kQ/(2 sqrt{2} L3)

Why does it follow that:

V(x) must be in the form V(x)= V(0) + @ x2 where V(0)= (sqrt{2} kQ)/L and the constant @ = kQ/ (sqrt{2} L 3) ?

Note that I haven't learned much yet on solving differential equations.
 
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Doesn't @ need to be kQ/ (4 sqrt {2} L 3)
 
Think about V(-x), i.e. is V(x) and odd or even function?
 

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