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Chapter 5: Q12RP (page 306)

In Problems 12 and 13, solve the related phase plane equation for the given system. Then sketch by hand several representative trajectories (with their flow arrows) and describe the stability of the critical points (i.e., compare with figure 5.12, page 267).

\({\bf{x' = y - 2,y' = 2 - x}}\)

Short Answer

Expert verified

The critical point is (2, 2) and the solution is\({{\bf{(y - 2)}}^{\bf{2}}}{\bf{ + (x - 2}}{{\bf{)}}^{\bf{2}}}{\bf{ = }}{{\bf{c}}^{\bf{2}}}\).

Step by step solution

01

Find the critical points

Here\({\bf{x' = y - 2,y' = 2 - x}}\)

For the critical points put the system equal to zero.

\(\begin{array}{c}{\bf{x' = 0}}\\{\bf{y - 2 = 0}}\\{\bf{y = 2}}\\{\bf{y' = 0}}\\{\bf{2 - x = 0}}\\{\bf{x = 2}}\end{array}\)

So, the critical point is (x, y) = (2, 2).

02

Solve for phase equation

The phase plane equation is:

\(\begin{array}{c}\frac{{{\bf{dy}}}}{{{\bf{dx}}}}{\bf{ = }}\frac{{{\bf{2 - x}}}}{{{\bf{y - 2}}}}\\\int {{\bf{(y - 2)dy}}} = \int {{\bf{(2 - x)dx}}} \\\frac{{{{\bf{y}}^{\bf{2}}}}}{{\bf{2}}}{\bf{ - 2y = 2x - }}\frac{{{{\bf{x}}^{\bf{2}}}}}{{\bf{2}}}{\bf{ + c}}\\{{\bf{(y - 2)}}^{\bf{2}}}{\bf{ + (x - 2}}{{\bf{)}}^{\bf{2}}}{\bf{ = 2c - 8}}\\{{\bf{(y - 2)}}^{\bf{2}}}{\bf{ + (x - 2}}{{\bf{)}}^{\bf{2}}}{\bf{ = }}{{\bf{c}}^{\bf{2}}}\end{array}\)

03

Step 3:Sketch

Therefore, this is the required result.

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Most popular questions from this chapter

Show that the Poincare map for equation (1) is not chaoticby showing that if\({\bf{(}}{{\bf{x}}_{\bf{o}}}{\bf{,}}{{\bf{\nu }}_{\bf{o}}}{\bf{)}}\)and\({\bf{(}}{{\bf{x}}^{\bf{*}}}_{\bf{o}}{\bf{,}}{{\bf{\nu }}^{\bf{*}}}_{\bf{o}}{\bf{)}}\)are two initial values that define the Poincare maps\({\bf{(}}{{\bf{x}}_{\bf{n}}}{\bf{,}}{{\bf{\nu }}_{\bf{n}}}{\bf{)}}\) and\({\bf{(}}{{\bf{x}}^{\bf{*}}}_{\bf{n}}{\bf{,}}{{\bf{\nu }}^{\bf{*}}}_{\bf{n}}{\bf{)}}\), respectively, using the recursive formulas in (3), then one can make the distance between\({\bf{(}}{{\bf{x}}_{\bf{n}}}{\bf{,}}{{\bf{\nu }}_{\bf{n}}}{\bf{)}}\)and\({\bf{(}}{{\bf{x}}^{\bf{*}}}_{\bf{n}}{\bf{,}}{{\bf{\nu }}^{\bf{*}}}_{\bf{n}}{\bf{)}}\)small by making the distance between\({\bf{(}}{{\bf{x}}_{\bf{o}}}{\bf{,}}{{\bf{\nu }}_{\bf{o}}}{\bf{)}}\) and \({\bf{(}}{{\bf{x}}^{\bf{*}}}_{\bf{o}}{\bf{,}}{{\bf{\nu }}^{\bf{*}}}_{\bf{o}}{\bf{)}}\)small. (Hint: Let \({\bf{(A,}}\phi {\bf{)}}\)and \({\bf{(}}{{\bf{A}}^{\bf{*}}}{\bf{,}}{\phi ^ * }{\bf{)}}\) be the polar coordinates of two points in the plane. From the law of cosines, it follows that the distance d between them is given by\({{\bf{d}}^{\bf{2}}}{\bf{ = (A - }}{{\bf{A}}^{\bf{*}}}{{\bf{)}}^{\bf{2}}}{\bf{ + 2A}}{{\bf{A}}^{\bf{*}}}{\bf{(1 - cos(}}\phi {\bf{ - }}{\phi ^ * }{\bf{))}}\).)

Use the Runge–Kutta algorithm for systems with h= 0.1 to approximate the solution to the initial value problem.

x'=yz;x(0)=0,y'=-xz;y(0)=1,z'=-xy2;z(0)=1,

At t=1.

In Problems 23 and 24, show that the given linear system is degenerate. In attempting to solve the system, determine whether it has no solutions or infinitely many solutions.

D-1x+D-1y=-3e-2t,D+2x+D+2y=3et

In Problem 36, if a small furnace that generates 1000 Btu/hr is placed in zone B, determine the coldest it would eventually get in zone B has a heat capacity of 2°F per thousand Btu.

In Problems 19–24, convert the given second-order equation into a first-order system by setting v=y’. Then find all the critical points in the yv-plane. Finally, sketch (by hand or software) the direction fields, and describe the stability of the critical points (i.e., compare with Figure 5.12).

d2xdt2-y=0
See all solutions

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