91Ó°ÊÓ

Solve the equation \(\dot{x}=x\) by Newton's method with the initial condition. \(\varphi(0)=1\)

Solve the difference equation \(\Delta^{3} \varphi=0\) with the initial condition \(\varphi(0)=\) \(0,(\Delta \varphi)(0)=0,\left(\Delta^{2} \varphi\right)(0)=2\) for \(t\) a multiple of the step size \(h=1\)

Knowing the components of the fields \(a\) and \(b\) in bome coordinate system, find the components of their commutator.

. Find the derivative of the solution \(\varphi\) of the equation \(\dot{x}=x^{2}+x \sin t\) with respect to the initial condition \(\varphi(0)=a\) for \(a=0\).

Let \(\left\\{g^{t}\right\\}\) be the phase flow of the field \(a\) and \(\left\\{h^{*}\right\\}\) the phase flow of the field \(\boldsymbol{b}\). Prove that the flows commute \(\left(g^{t} h^{\prime} \equiv h^{\circ} g^{t}\right)\) if and only if the commutator of the fields is zero.

Study the self-oscillating modes of the motion of a pendulum with small friction under the action of a constant torque \(M\) : $$ \tilde{x}+\sin x+\varepsilon \dot{x}=M $$

Construct the equation for the evolution of the velocity field of a medium of noninteracting particles in a force field with force \(F(x)\) at the point \(x\).

Find the derivative of the solution of the pendulum equation \(\bar{\theta}=-\sin \theta\). with initial condition \(\theta(0)=a, \dot{\theta}(0)=0\) with respect to \(a\) at \(a=0\).

For which natural numbers \(k\) can all the solutions of the equation \(\bar{x}=x^{k}\) be extended infinitely far?

Construct the equation for the evolution of the velocity field of a medium of noninteracting particles in a force field with force \(F(x)\) at the point \(x\).