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There are a few posts on this forum that deals with the transformation of non-autonomous differential equations into autonomous differential equations, in particular: Transforming differential equations into autonomous first-order systems and When can a non-autonomous system NOT be re-written as an autonomous system?.

Mainly I'm currently tasked to transform the Bessel Equation $$ t^2u^{''}(t) + tu^{'}(t) + (t^2 - \alpha^2)u(t) = 0 $$ into a system of autonomous differential equations of first order. The transform to first order is quite simple by defining $z_1(t) = u(t)$ and $z_2(t) = u'(t), yielding \begin{align*} z_1^{'}(t) &= z_2(t) \ \end{align*} \begin{align*} z_2(t) &= -\frac{1}{t}z_2(t) - (1 - \frac{\alpha^2}{t^2})z_1(t) \ \end{align*}

To make this system autonomous, we can introduce some parametrization $s(t)$ with $s^{'}(t) = 1$, expanding the above system to \begin{align*} z_1^{'}(s) &= z_2(s) \\ \end{align*} \begin{align*} z_2(s) &= -\frac{1}{s}z_2(s) - (1 - \frac{\alpha^2}{s^2})z_1(s) \\ \end{align*} \begin{align*} s^{'}(t) = 1 \end{align*} However I don't see how this system is in any way easier to analyse or to solve. The equation $z_2(s) = -\frac{1}{s}z_2(s) - (1 - \frac{\alpha^2}{s^2})z_1(s)$ didn't get any easier to tackle with the common solution methods and I'm just kind of stumped why this step is useful (not just for the Bessel Equation but also for the general case).

Zedssad
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1 Answers1

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I've found at least a partial answer when studying flows of ODEs with this result:

Let $D \subset \mathbb{R} \times \mathbb{R}^n$ be some open subset an $w: D \rightarrow \mathbb{R}^n, t \mapsto w(t,x)$ a time-dependent $C^{\infty}$ vector field inducing some non-autonomous ODE $\dot{x} = w(t,x)$. Now let $$v: D \rightarrow \mathbb{R}^{n+1}, \tilde{x} = (x_0,x) \mapsto (1, w(x_0,x))$$ be the corresponding autonomous system. Now define the maximal local flow corresponding to $v$ as $$\Phi : A = \bigcup_{\tilde{x}\in D} (I^-_\tilde{x},I^+_\tilde{x}) \times \{ x\} \rightarrow \mathbb{R}^{n+1}$$ Then for any $(\tau,\xi) \in D$ the unique maximal solution to the IVP \begin{align*} \dot x &= w(t,x) \\ x(\tau) &= \xi \end{align*} is given by $$x: (I^-_\tilde{(\tau,\xi)} + \tau,I^+_\tilde{(\tau,\xi)} + \tau) \rightarrow \mathbb{R}^{n}, t \mapsto \text{pr} \circ \Phi(t - \tau, (\tau, \xi))$$ where $\text{pr} : \mathbb{R}^{n+1} = \mathbb{R} \times \mathbb{R}^{n} \mapsto \mathbb{R}^{n}, (x_0,x) \mapsto x$ is the projection onto the second to last coordinates.

This result is quite easy to prove and now establishes a connection between the autonomous and non-autonomous equation.

In particular, some powerful results for flows corresponding to autonomous vector-fields (e.g. if the flow for some autonomous equation exists at one $t_0 > 0$ for all points $x$, then it exists for all $t \in [0, \infty)$) can be transferred to the non-autonomous system by projecting the flow.

Zedssad
  • 669