The real and imaginary parts of the general solution are two (linearly independent) real solutions.
I've put the ODE into a system.
We have:
which has Eigenvalues:
and Eigenvectors:
,
General Solution is
But this is where I am getting lost, I know that when the eigenvalues are complex I need to involve sin and cos, which is fine for a 1st order ODE but am getting a little stuck with this 2nd-order ODE.
Also, I need to find the solution in Real Form. i.e.
Thanks in advance to anyone who can help.
I was wondering if someone could check my solution. If its not correct, I think it could be close.....
General Solution:
(Complex Form)
Note: Don't confuse and with and
where
and
and
(I'm sorry if this part is confusing, but when I tried to write one piece of latex code for this I got an error message saying something like 'Latex Error too big')
....
.....
(Real Form)
whew....(wipes brow)