Euler's Formula

Euler's formula is used in Differential Equations to take complex solutions and turn them into real-valued solutions. An example of a complex solution is a solution that has a complex eigenvalue, such as
     `bb(Y) (t) = e^((-2 + 3i)t) ((i), (1))` (Blanchard, Devaney, Hall, 293).

As you can see, `e^((-2 + 3i)t)` is of the form `e^(a + ib),` with a factored `t`, which is the same as the form used in Euler's formula above.
To take the complex solution and turn it into a real solution, you use Euler's formula where  `e^((-2 + 3i)t)` becomes

`e^(-2t)(cos(3t) + i sin(3t))`, which is then multiplied by the eigenvector `((i), (1))` to lead us to 

     `bb(Y) (t) = e^((-2 + 3i)t) ((i), (1))=e^(-2t)*((-sin(3t)), (cos(3t)))+e^(-2t)*i((cos(3t)),(sin(3t))).`

Thus,
    `e^(-2t)*((-sin(3t)), (cos(3t))) " and  " e^(-2t)*((cos(3t)),(sin(3t)))`

are the two real, linearly independent solutions to the system, so the general solution is any linear combination of them. So the final answer becomes
     `bb(Y) (t) =c_(1)((-e^(-2t)sin(3t)), (e^(-2t)cos(3t)))+c_(2)((e^(-2t)cos(3t)),(e^(-2t)sin(3t))),`

and if given initial conditions we could now use them to solve for the constants `c_(1)` and `c_(2).`

Sources

Blanchard, Paul, Robert L. Devaney, and Glen R. Hall. Differential Equations. 3rd ed. Belmont, CA: Thomson Brooks/Cole, 2006. Print.