What you are saying is that you want to solve the differential equation y''''+ y= 0. That is a linear equation with constant coefficients. If you try a solution of the form , the fourth derivative is so the equation becomes . That's the "characteristic equation" for this differential equation. It is equivalent to which has four complex solutions, two pairs of complex conjugates. To write the solution to the differential equation in terms of real functions, use [tex]e^{a+ bi}= e^ae^{bi}= e^a(cos(b)+ i sin(b))