I have the DE:
If I convert this to a system of 1st order ODES is it just:
How about if the DE is:
and the same IC's.
Clearly this has the imaginary roots and
Compute and sub in initial conditions.
The second question can be computed by method of comparing co-efficients (we can also use variation of paramters but co-efficients is easiast here).
For the general solution is the solution from question 1 plus the particular solution.
For the particular solution,
Subbing into the equation we get,
That's very clever, guys, but it doesn't actually answer the question asked, does it?
Royal Flush, yes, if you let and , then we have immediately that and then so that so that
And your initial conditions are and .
This can also be written as the "matrix" equation:
subject to the initial condition .
In the same way, the equation y"+ y= sin(2t) becomes the pair of equations and with the same intial conditions.
That also can be written as a "matrix" equation:
with initial condition .
Of course, the eigenvalues of the matrix are i and -i which lead to the complex exponentials (sine and cosine) solutions Prove It and AllenCuz show.