Second order differential equation, complex roots

[dbag]

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 e^itheta= 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?

 

[Deleted member 4993]

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 e^itheta= 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]

Thank you very much