Transforming differential equations into autonomous first-order systems

Question

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

Answer 1

You use the time as an additional component with derivative $1$. $$ \frac{d}{dt}\pmatrix{x_0 \\ x_1 \\ x_2}=\pmatrix{1 \\ x_2 \\ x_0\sin(x_2)+x_1} $$

Answer 2

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.