Prove that the solutions to an ODE are non-periodic oscillating. I have the ODE $$\ddot{x}+ \dot{x}+k^2x=0$$
where the dot means derivative with respect to $t$ ($x$ is a function of $t$) and $k$ is a constant such that $k>\frac{1}{2}$
prove that the solutions to the ODE are non-periodic and oscillating.
What I tried is the following.
I solved the characteristic polynomial equation $$\lambda^2+\lambda+k^2=0$$
which gave me $$ \lambda_{1,2} = \frac{-1 \pm 2 \sqrt{k^2-\frac{1}{4}}}{2} $$
Now since $k>\frac{1}{2}$
the solutions are real so the ODE has no periodic solutions, but I can't understand how to prove that the solutions oscillate, what does it even mean for the solutions to oscillate?
 A: Given the equation
$\ddot x + \dot x + k^2x = 0, \tag 1$
the characteristic equation is indeed
$\lambda^2 + \lambda + k^2 = 0, \tag 2$
but the "quadratic formula" used to find $\lambda_{1,2}$ is misstated; it should read
$\lambda_{1,2} = \dfrac{-1 \pm \sqrt{1 - 4k^2}}{2}; \tag 3$
from this we see that when $k > 1/2$ the values of $\lambda_{1,2}$ are complex conjugate numbers with real part $-1/2$, hence the solutions exponentially decay to $0$; however the presence of a non-vanishing imaginary component in the $\lambda_{1,2}$ forces the solutions to venture
back and forth across $0$ whilst they decay; thus the solutions are non-periodic and oscillating.
We can in fact write down explicit solutions to (1); they serve to reify and illustrate the words of the preceding paragraph.  Writing (3) in the form
$\lambda_{1, 2} = -\dfrac{1}{2} \pm i \dfrac{1}{2}\sqrt{4k^2 - 1}, \tag 4$
we may express two relatively simple solution as
$x_\pm(t) = \exp \left (\lambda_{1,2} t\right ) = \exp \left (\left (-\dfrac{1}{2} \pm i \dfrac{1}{2}\sqrt{4k^2 - 1} \right)  t\right )$ =
$\exp \left (-\dfrac{1}{2}  t\right )\exp \left (\pm i \dfrac{1}{2}\sqrt{4k^2 - 1} t\right )$
$= \exp \left (-\dfrac{1}{2}  t\right )\left (\cos \left (\dfrac{1}{2}\sqrt{4k^2 - 1} t \right ) \pm i \sin \left (\dfrac{1}{2}\sqrt{4k^2 - 1} t \right )\right). \tag 5$
Since (1) is a real differential equation, the real and imaginary parts of $x_\pm(t)$ are individually solutions, thus for example
$x(t) =  \exp \left (-\dfrac{1}{2}  t\right )\left (\cos \left (\dfrac{1}{2}\sqrt{4k^2 - 1} t \right ) \right) \tag 6$
is a solution which exhibits both exponential decay and oscillation.
