Toronto Math Forum
MAT2442018F => MAT244Lectures & Home Assignments => Topic started by: Xin Wen on November 24, 2018, 04:57:45 PM

Can any one give me the idea about 9.2 problem set Q21?
Given that $x=\phi(t)$, $y=\psi(t)$ is a solution of the autonomous system
\begin{equation}
\frac{dx}{dt} = F(x,y), \frac{dy}{dt}=G(x,y)
\end{equation}
for $\alpha < t < \beta$, show that $x=\Phi(t)=\phi(ts), y=\Psi(t)=\psi(ts)$ is a solution for $\alpha + s < t < \beta +s $ for any real number s.

Since $x=\phi(t)$, $y=\psi(t)$ is a solution of the autonomous system
\begin{equation}
\frac{dx}{dt} = F(x,y), \frac{dy}{dt}=G(x,y)
\end{equation}
for $\alpha < t < \beta$.
Then functions $\Phi(t)=\phi(ts)$ and $\Psi(t)=\psi(ts)$ indicates same function $x=\phi(t)$ and $y=\psi(t)$ respectively (except time shift by a real number s)
Base on the definition of Autonomous system (on textbook page 509) "an autonomous system is one whose configuration, including physical parameters and external forces or effects, is independent of time."
Thus, $x = \Phi(t)=\phi(ts)$, $y=\Psi(t)=\psi(ts)$ is a solution for $\alpha +s < t < \beta$ +s for any real number s.