Thread: Using complex numbers for 2nd order DE

I can change this equation, using Eulers method, and get the answers using and , but using complex numbers is apparently faster and we have to do it this way for my professor...so if you could help me figure this out I would appreciate it.



HERE IS WHAT I DID:
First, I'll solve the homogeneous case to find the complementary solution and the roots.

or


Now, to find the solution , where is the particular solution to the non-homogeneous equation, I need help. I guessed a few functions but none seemed to work out.

Here is what Ive tried:
Since ...well the real part I think
and
is a root of the homogenous equatoin, so I need multiply my guess for by

From here I cannot figure out what function I need to use...

Originally Posted by snaes

I can change this equation, using Eulers method, and get the answers using and , but using complex numbers is apparently faster and we have to do it this way for my professor...so if you could help me figure this out I would appreciate it.



HERE IS WHAT I DID:
First, I'll solve the homogeneous case to find the complementary solution and the roots.

or


Now, to find the solution , where is the particular solution to the non-homogeneous equation, I need help. I guessed a few functions but none seemed to work out.

Here is what Ive tried:
Since ...well the real part I think
and
is a root of the homogenous equatoin, so I need multiply my guess for by

From here I cannot figure out what function I need to use...

Since the complementary solution involves the RHS, you will need to guess a particular solution of the form

.

Thanks, but is there any way to convert this into a "complex" valued expression. I need to use something like: .

[edit] for myhomework we are supposed to do each problem with real valued (sine and cosine) which i can do AND using some kind of form, which i cannot do. Any help would be great.