3.53 Show that each of the following systems has no cycles:(i)I – y + ry²,61FIGURE 40. An invariant region for the model of glycolysis.(ii)1+ry,2.1² + x²y?%3D(iii)= -I- y+2x²+ y²,y'%3D

3.53 Show that each of the following systems has no cycles: (i) I = 1-y+ry², 61 FIGURE 40. An invariant region for the model of glycolysis. r' = 1+ ry, 2.12 + x?y? %3D (iii) r = -1- y+2x² + y², %3D %3D

3.53 Show that each of the following systems has no cycles: (i) I = 1-y+ry², 61 FIGURE 40. An invariant region for the model of glycolysis. r' = 1+ ry, 2.12 + x?y? %3D (iii) r = -1- y+2x² + y², %3D %3D

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter9: Systems Of Equations And Inequalities

Section: Chapter Questions

Problem 12T

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Question

3.53 Show that each of the following systems has no cycles:
(i)
I – y + ry²,
61
FIGURE 40. An invariant region for the model of glycolysis.
(ii)
1+ry,
2.1² + x²y?
%3D
(iii)
= -I- y+2x²+ y²,
y'
%3D

Expert Solution

Step 1

Consider the given information,

Here, Benderson's criteria is defined as,

If f'(x) and g'(y) are continuous region on R and $\frac{\partial f}{\partial x} + \frac{\partial g}{\partial y} \neq 0$ .

Then the region $x' = f (x, y) and y' = g (x, y)$ has no cycle. This implies that there is no closed trajectory in R.

Step 2

(1)

Consider the given equation.

$\begin{array}{l} x' = x - y + x y^{2} = f (x, y) \\ y' = x = g (x, y) \end{array}$

Now, find the derivative with respect x for f and with respect y for g.

$\begin{array}{l} f_{x} = 1 + y^{2} \\ g_{y} = 0 \end{array}$

So,

$\begin{array}{rcl} \frac{\partial f}{\partial x} + \frac{\partial g}{\partial y} & = & 1 + y^{2} + 0 \\ \neq & 0 \end{array}$

Thus, the system have no cycle.

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