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.