4th order linear ordinary differential equation

Question

While I was solving an integral using Feynman Integration, I came across the following differential equation:

$$y’’’’-y’’+y=0$$

I tried substituting $y$ with an exponential function which failed. Can someone else show me how to solve it?

Answer 1

$$λ^4−λ^2+1$$ can be completed to a square by variation of the middle term, making it more negative in the process $$=(λ^2+1)^2−3λ^2=(λ^2+\sqrt3λ+1)(λ^2-\sqrt3λ+1)$$ Now one can apply the usual solution formulas for quadratic equations (with real coefficients but complex roots).