I am self-studying ODEs. My professor suggested Arnold's book. I am studying Rectifications. and it is not clear for me at all.
Do you know of any link which explains what rectification means and what are the steps to take? This is an example question which I have no idea what it is asking for.
Rectify the integral curves of the equation $\dot x=x+\cos t$
Linearize at $(0,1)$ ...( I have seen an example of linearization using Jacobean matrix and discussing the stability of points and so on)
From the given equation,
$$\dot x(0)=x(0)+\cos(0)=2$$ and the linearization is
$$x\approx1+2t.$$
To rectify, you compute the arc length as
$$s(t)=\int_0^t\sqrt{1+\dot x^2}\,dt\approx\int_0^t\sqrt{5}\,dt=\sqrt 5t.$$
You can obtain higher order approximations using the Taylor development of $s$:
$$s(0)=0,$$
$$\dot s(0)=\left.\sqrt{1+\dot x^2}\right|_{t=0}=\sqrt5,$$
$$\ddot s(0)=\left.\frac{\dot x\ddot x}{\dot s}\right|_{t=0}=\frac{2(2-\sin 0)}{\sqrt 5}=\frac4{\sqrt5},$$
$$\dddot s(0)=\left.\frac{(\ddot x^2+\dot x\dddot x)\dot s-\dot x\ddot x\ddot s}{\dot s^2}\right|_{t=0}=\frac{(2^2+2(2-\cos1))\sqrt5-2\cdot2\dfrac4{\sqrt5}}5=\frac{14}{5\sqrt5},$$
$$\cdots$$
and
$$s(t)\approx \sqrt5t+\frac{2t^2}{\sqrt5}+\frac{7t^3}{15\sqrt5}+\cdots$$
The computation of the next terms is a little tedious.