write as

x '' + x = 0

Assume solution of the form x = e^(rt)

Then plugging back into the original we obtain:

r^2e^(rt) + e^(rt) = 0

r^2 +1 = 0

This is the characteristic equation the solution is r =+i

The solutions are of the form e^(it) (you only need one of the complex solutions)

Using Euler's idenity e^(it) = cos(t) + i sin(t)

Then the real part: cos(t) and the imaginary part : sin(t) are the solutions

The general solution is x = Acos(t) + Bsin(t) which is easilyu verified to solve x '' + x = 0