I was given this non-homogeneous second order differential equation:
$\dfrac{d^2y}{dx^2} + y = sec(x)$
In order to solve this, I need to find the general solution to the homogeneous part by finding the characteristic equation which is:
$\lambda^2 + 1 = 0$
$\lambda^2 = -1 $
$\lambda = \pm i $
According to my understanding, based on the roots from characteristic equation , the general solution to homogeneous part should be
$y_{homogeneous} = Ae^{ix} + Be^{-ix}$
But Instead,In my text, it was written as this:
$y_{homogeneous} = Acos(x) + Bsin(x)$
I am aware that I could use Euler's formula to convert my exponential terms into sine and cosine but even then I would not be able to get the solution given in the text.
What am i missing?