What is the general solution to y''''+y=0? I get to the characteristic equation (r^2+i)(r^2-i)=0. Using the Wolframalpha website to find the roots I get r1 = +(-1)^1/4, r2 = -(-1)^1/4, r3 = +(-1)^3/4 and r4 = -(-1)^3/4. I have a feeling its going to be in some form of sine & cosine (have done 2nd order ODE's) but am at a loss to get to the exact solution for this 4th order ODE. I would appreciate any guidance!

Question

What is the general solution to y''''+y=0?  I get to the characteristic equation (r^2+i)(r^2-i)=0.  Using the Wolframalpha website to find the roots I get r1 = +(-1)^1/4, r2 = -(-1)^1/4, r3 = +(-1)^3/4 and r4 = -(-1)^3/4.  I have a feeling its going to be in some form of sine & cosine (have done 2nd order ODE's) but am at a loss to get to the exact solution for this 4th order ODE.  I would appreciate any guidance!