Finding the homogeneous part of general solution to second order non-homogeneous differential equation

Question

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?

Answer 1

You're on the right path. Like you said, why don't you use Euler's formula to turn your exponentials into sines and cosines and then collect like terms, remembering that

(A+B)

and

(A-B)i

are just constants, so can be replaced with another letter of your choosing.