A few weeks ago, I was asked the following in a homework assignment
Study the system $\dot{x}(t)=-\epsilon x(t)\cos(t)$ by the method of averaging and compare this to the exact solution. My exact solution is
$$x(t)=x(0)\exp{[-\epsilon \sin(t)}]$$
This is very straight forward to show (this is a separable ODE). But I'm not too sure about the method of averaging.
There is a section in Arnold's Mechanics on this..... which I don't understand. Any ideas greatly appreciated.
A partial result is that the solution should look like the following:
$$x(t) \approx x(0)(1-\epsilon t)$$ This can be obtained by Taylor expanding the exact solution of the exponential, throwing out terms squared or higher, and approximating $\sin(t)$ by t.
Here is what Arnold's book says:
Let $I, \varphi$ be action-angle variables in an integrable ('nonperturbed') system with Hamiltonian function $H_0(I)$: $$\dot{I}=0 ;\dot{\varphi}=\omega(I);\omega(I)=\frac{\partial H_0}{\partial I}$$
As the nearby "perturbed" system we take the system
$$\dot{\varphi}=\omega(I)+\epsilon f(I,\varphi); I=\epsilon g(I,\varphi) : \epsilon<<1$$
The $ averaging$ $principle$ $for$ $the$ $system$ consists of its replacement by another system, called the averaged system:
$$\dot{J}=\epsilon \hat{g}(J); \hat{g}(J)=(2\pi)^{-k} \int_0^{2k}\cdots\int_0^{2k} g(J,\varphi) d\varphi _1,... d\varphi _k$$
I have a little bit of an idea how to do this... but not too clear.
A better source to look for basics of the averaging method is the book by F. Verhulst "Nonlinear Differential Equations and Dynamical Systems".
In short, if you have an equation of the form $$ \dot x=\varepsilon f(t,x)+\varepsilon ^2g(t,x,\varepsilon), $$ then, considering $$ f^0(y)=\frac{1}{T}\int_0^T f(t,y)dt, $$ it is possible to prove that solution to $$ \dot y=\varepsilon f^0(y) $$ is $\varepsilon$-close to solution $x(t)$ on the time scale $1/\varepsilon$.