General solution of the ordinary differential equation $(D^4+D^2+1)y=0$

Solution 1:

Your equation is $$x^4+x^2+1=0$$ Let $y=x^2$, then we have that: $$y^2+y+1=0$$ And the solution is $y=-\frac{1}{2}\pm\frac{\sqrt{3}}{2}i$, or in exponential form: $y_1=e^{\frac{2}{3}i\pi}$ and $y_2=e^{-\frac{2}{3}i\pi}$. And from $x^2=y$, we get that $x_{12}=\exp\left(\frac{\frac{2}{3}i\pi+2ni \pi}{2}\right)$ and $x_{34}=\exp\left(\frac{-\frac{2}{3}i\pi+2ni \pi}{2}\right)$ for $n=0$ and $n=1$. So the roots are: $$x_1=\exp\left(\frac{1}{3}i \pi\right)$$ $$x_2=\exp\left(\frac{4}{3}i \pi\right)$$ $$x_3=\exp\left(-\frac{1}{3}i \pi\right)=\exp\left(\frac{5}{3}i \pi\right)$$ $$x_4=\exp\left(\frac{2}{3}i \pi\right)$$