Problem 1.7 in G.Teschl ODE and Dynamical Systems asks me to transform the following differential equation into autonomous first-order system:
$\ddot x = t\sin(\dot x) +x$
Transforming the ODE to a system is in this case easy, but whats the usual technique to transform it to an AUTONOMOUS system?
Thanks so much <3
Best Answer
Given that
$\ddot x = t\sin \dot x + x, \tag 1$
we may set
$y = \dot x, \tag 2$
then
$\dot y = \ddot x, \tag 3$
and we have
$\dot y = t\sin y + x; \tag 4$
we also need to replace the independent variable $t$ with yet a third (in addition to $x$ and $y$) dependent variable we shall call $z$, which obeys
$\dot z = 1; \tag 5$
this equation implies that
$z = t + c, \; c \in \Bbb R; \tag 6$
we need to ensure that $c = 0$; this may be effected by setting an initial value for $z$, to wit:
$z(t_0) = t_0; \tag 7$
then
$c = 0, \tag 8$
and
$z = t; \tag 9$
thus the sought-for autonomous system may be written
$\dot x = y, \tag{10}$
$\dot y = z\sin y + x, \tag{11}$
$\dot z = 1, \; z(t_0) = t_0; \tag{12}$
note we also need to specify $x(t_0)$ and $y(t_0)$ to obtain a unique solution.