Averaging estimate on $[0,100]$ for the scalar equation $x^{\mathrm{\prime }}=\epsilon \cos t$

Given: The data above and the scalar ODE $x^{\mathrm{\prime }}=\epsilon \cos t$, $x(0)=0$.

For this $f$, $\bar{f}(x)=\frac{1}{2\pi }{\int }_0^{2\pi }\cos sds=0$ and $g(t,x)={\int }_0^t\cos sds=\sin t$ on $[0,2\pi ]$, extended periodically; this is the construction of Periodic averaging data: the average field, the oscillatory part and the periodic deviation, evaluated with the fundamental theorem for continuous integrands (Every continuous function on an interval has a primitive; two primitives differ by a constant; and ${\int }_a^{b}f=G(b)-G(a)$ for any primitive $G$, The derivatives of sine and cosine are cosine and minus sine); the endpoint values $\sin 0=\sin (2\pi )=0$ and periodicity used here are supplied by [L4]. [given]

The averaging theorem (Averaging for fast periodic oscillations: order-$\epsilon$ agreement on the $O(1\mathrm{/}\epsilon )$ time scale): under its hypotheses with a solution $y:[0,A\mathrm{/}\epsilon ]\to \bar{B}(x_0,R-\rho )$ of the averaged system and $\epsilon K<\rho$, the solution $x$ of $x^{\mathrm{\prime }}=\epsilon f(t,x)$, $x(0)=x_0$ exists on $[0,A\mathrm{/}\epsilon ]$ and satisfies $\mathrm{\parallel }x(t)-y(t){\mathrm{\parallel }}_2\le \epsilon K$ there, with $K=e^{LA}[B+(LB+B_{1}M)A]$, $B=2MP$, $B_1=2LP$. [given]

The definition $\pi =2\gamma$ and theorem $0<\gamma <2$ give $0<\pi <4$ (Pi as twice the smallest positive zero of cosine, Cosine has a smallest positive zero, lying strictly between zero and two).

Sine and cosine are $2\pi$-periodic, and $\sin (2\pi )=\sin 0=0$ by their zero set (The zero sets of sine and cosine and the least positive common period 2 pi). Their ranges are $[-1,1]$ (Signs, monotonicity intervals, and ranges of sine and cosine); in particular $\mathrm{\mid }\cos t\mathrm{\mid },\mathrm{\mid }\sin t\mathrm{\mid }\le 1$. Their differentiability in [L1] gives continuity.