Hailie Blevins

2022-06-22

Given the system of differential equations ${x}^{\prime}=2x+{y}^{3}$ and ${y}^{\prime}=-y$ i found the flow

${\varphi}_{t}(x,y)=(({x}_{0}+1/5{y}_{0}^{3}){e}^{2t}-1/5{y}_{0}^{3}{e}^{-3t},{y}_{0}{e}^{-t})$

${\varphi}_{t}(x,y)=(({x}_{0}+1/5{y}_{0}^{3}){e}^{2t}-1/5{y}_{0}^{3}{e}^{-3t},{y}_{0}{e}^{-t})$

pheniankang

Beginner2022-06-23Added 22 answers

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