# Second order differential equation, complex roots

#### dbag

##### New member
Hello

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 eitheta= cos(theta) + isin(theta)

and that comes out to:
y = e0(a1*cos(wt)+isin(wt)+a2cos(-wt)+isin(-wt))
or
y = e0(a1+a2)*cos(wt)+(a1-a2)i*sin(wt))

We let a1 + a2 = C1 and a1-a2 = C2

The answer then comes out to:
Y = C1cos(wt)+C2sin(wt) ?

Is that right?

#### Subhotosh Khan

##### Super Moderator
Staff member
Hello

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 eitheta= cos(theta) + isin(theta)

and that comes out to:
y = e0(a1*cos(wt)+isin(wt)+a2cos(-wt)+isin(-wt))
or
y = e0(a1+a2)*cos(wt)+(a1-a2)i*sin(wt))

We let a1 + a2 = C1 and a1-a2 = C2

The answer then comes out to:
Y = C1cos(wt)+C2sin(wt) ?

Is that right?
Looks good to me. Thanks for sharing your work and thoughts.

#### dbag

##### New member
Thank you very much