I am given $y^{(8)}+y^{(6)}+y^{(4)}+y^{(2)}+ay=0$ where $a$ is a real number. The question asks for which values of $a$ does the equation have a real-valued solution which is never $0$.
Convert to characteristic equation: $r^8 + r^6 + r^4 + r^2 + a=0$
Our solution will be of the form $y(t)=C_1e^{r_1t}+...+C_8e^{r_8t}$. In order for this to never be zero, we need atleast one $C_i \neq 0$ and no two terms that are the same but opposite signs. I am struggling with manipulating the characteristic equation to figure out how $r$ and $a$ depend on each other. I tried factoring and rewriting, but I couldn't find any useful information. Any help is appreciated. Thank you.