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I am reading Chicone's book and it mentions that you can reduce an $n$th order differential equation into a system of first order equations, and gives an example. Nevertheless, there's no example of how you convert a nonautonomous differential equation to an autonomous one. How does this work, say with $x' = x + t$?

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You fix $y = t$ and $x = x$ so that $y' = 1$ and you have the system \begin{align} x' &= x + y, \\ y' &= 1, \end{align} with initial condition $y(0) = 0$, and $x(0) = x_0$.

In general, if $y \in \mathbb R^d$, $f\colon \mathbb R \times \mathbb R^d \to \mathbb R^d$, and you have the ODE $y' = f(t, y)$ with IC $y(0) = y_0 \in \mathbb R^d$, you can make the change of variable $Y = (y, t)$ and $F(Y) = (f(Y_{d+1}, Y_{1:d}), 1)$ to get the autonomous ODE $Y' = F(Y)$ with initial condition $Y(0) = (y_0, 0)$.

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