I am trying to get the general solution to a differential equation. The roots to the equation are complex roots w(omega)i and -wi

solution 1 is then e

^{(wi)t}and solution 2 is e

^{(-wi)t}

Then applying the Euler's rule e

^{itheta}= cos(theta) + isin(theta)

and that comes out to:

y = e

^{0}(a

_{1}*cos(wt)+isin(wt)+a

_{2}cos(-wt)+isin(-wt))

or

y = e

^{0}(a

_{1}+a

_{2})*cos(wt)+(a

_{1}-a

_{2})i*sin(wt))

We let a

_{1}+ a

_{2}= C

_{1}and a

_{1}-a

_{2}= C

_{2}

The answer then comes out to:

Y = C

_{1}cos(wt)+C

_{2}sin(wt) ?

Is that right?