Hailie Blevins

2022-06-22

Given the system of differential equations ${x}^{\prime }=2x+{y}^{3}$ and ${y}^{\prime }=-y$ i found the flow
${\varphi }_{t}\left(x,y\right)=\left(\left({x}_{0}+1/5{y}_{0}^{3}\right){e}^{2t}-1/5{y}_{0}^{3}{e}^{-3t},{y}_{0}{e}^{-t}\right)$

pheniankang

Since $y={y}_{0}{e}^{-t}$, there are no periodic solutions. If a solution had period $T>0$, then ${y}_{0}^{-\left(t+T\right)}={y}_{0}{e}^{-t}$, clearly impossible unless ${y}_{0}=0$. If ${y}_{0}=0$, $x={x}_{0}{e}^{2t}$ and a similar argument applies.