Chapter 2, Problem 1RE

To determine

To find:one solution of the given system.

Answer to Problem 1RE

The one of the solutions of the system is $x=0,y=0$ .

Explanation of Solution

Given information:

The system of equations are $\frac{dx}{dt}=\left|x\right|\sin y$ and $\frac{dy}{dt}=\left|y\right|\cos x$ .

Formulas used:

• Atequilibrium, all the rates are equal to zero. The equilibrium point is alsosolution of the system.

Calculation:

Solve for equilibrium points.

That is solve for $\frac{dx}{dt}=0$ and $\frac{dy}{dt}=0$

$\begin{array}{l}\frac{dx}{dy}=0\\ \Rightarrow \left|x\right|\sin y=0\\ \Rightarrow \left|x\right|=0\text{ or }\sin y=0\\ \Rightarrow x=0\text{ or }y=\mathrm{0......}(1)\end{array}$

Similarly,

  $\begin{array}{l}\frac{dy}{dx}=0\\ \Rightarrow \left|y\right|\cos x=0\\ \Rightarrow \left|y\right|=0\text{ or }\cos x=0\\ \Rightarrow y=0\text{ or }x=(\pi /2)+2\pi n,(3\pi /2)+2\pi n\end{array}$

Hence, one solution of the system is $x=0,y=0$ .