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**mathstudent1992** Hi, I am doing a problem and need help... I am really confused :/

Well, the problem says

"Show that the boundary conditions for $\displaystyle zZ''(z)+Z'(z)+v^2Z(z)$=0 are $\displaystyle |Z(0)|<\infty$ and $\displaystyle Z(L)=0$ (we know that $\displaystyle v>0$)."

$\displaystyle Z(z)$ comes from $\displaystyle y(z,t)=Z(z)T(t)$. This problem is about the dynamic behaviour of a hanging cable, so $\displaystyle L$ is the length of the cable and when $\displaystyle z=L$ it means that $\displaystyle z$ is at the highest point where the cable is hanging from, and $\displaystyle z=0$ is the end point, so $\displaystyle |Z(0)| \le L < \infty$. However, that's something that the problem tells us, so I cannot just say it is like that, I need to prove it.

So well, my first thought was to solve the differential equation, but I got to this point

$\displaystyle \lambda=\frac{-1 \pm \sqrt{1-4zv^2}}{2z}$

I know that $\displaystyle z$ cannot be zero, so $\displaystyle Z(0)$ would not be a solution, but that does not prove why $\displaystyle |Z(0)|<\infty$. So I guess I shouldn't be solving the differential equation.

Thanks a lot in advance for any help you can give me!