Question

.

Answer 1

Is that supposed to say \(\Large \dfrac {d^2n }{dx^2}=\dfrac {n}{L^2}\)?

Answer 2

yes

Answer 3

The second derivative of this currently is telling you, intuitively, that the second derivative of a function gives back the same function divided by a constant. There are two possibilities that I see, sin/cos, or e^x.

Answer 4

I haven't done diff. eq. in a while so all I can do is kinda trial and error here.

\(n(x)=Ae^{bx}\) I think is one.
The other would be
\(n(x)=Asin(kx)+Bcos(kx)\).

Not sure if the exponential one is correct.

Next, take the two functions, take the second derivative, and see which one fits.