FD2010 said:
I am terribly sorry but I wound up writing the original expression wrong. I was so hasty for help I had an error in the equation (I left out a u!). Here is what I am trying to solve (new image).
Any help is sincerely appreciated!
Nick
I don't see how you're getting all that unless the constants are eliminating the complex components of the solution. For starters, it's just one derivative so you could replace it with just an ODE.
[tex]u''-ku=0[/tex]
and keep in mind the i is just a constant and as long as you use correct complex-value arithmetic, it's just like real variables. So the solution to that is:
[tex]u(y)=c_1 e^{\sqrt{k}y}+c_2 e^{-\sqrt{k}y}[/tex]
however, that expression is already (implicitly) taking out the two values of the multi-valued square root. Now in the case of [itex]\sqrt{i}[/itex], it's best to explicitly evaluate the square root of i:
[tex]\sqrt{\frac{i\omega}{\nu}}=\sqrt{\omega/\nu}e^{\pi i/4}{, \sqrt{\omega/\nu}e^{-3\pi i/4}}[/tex]
then the solution with the i would be:
[tex]u(y)=c_1\text{exp}\left(y\sqrt{\omega/\nu}e^{\pi i/4}\right)+c_2\text{exp}\left(y\sqrt{\omega/\nu}e^{-3\pi i/4}\right)[/tex]
And if it was indeed a partial of say u(y,t), then the constants of integration would be functions of t and the solutions would be:
[tex]u(y,t)=c_1(t)\text{exp}\left(y\sqrt{\omega/\nu}e^{\pi i/4}\right)+c_2(t)\text{exp}\left(y \sqrt{\omega/\nu}e^{-3\pi i/4}\right)[/tex]
