You have applied Euler's method to solve the homogeneous with constant coefficeints.
The common solution of the equation can be expressed like this:
y = C1 e rx + C2 e -rx (1)
You have found the right equation for r after substituting (1) into you differential equation. Use
r = 2i and r = -2i
to come up with final solution:
y = C1 e 2ri + C2 e - 2ri
This solution can be expressed in terms of trigonometric functions, but coefficients C1 and C2 have to be found from boundary conditions which are not given here.