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Chapter 5: Q12E (page 271)

In Problems 11–14, solve the related phase plane differential equation for the given system. Then sketch by hand several representative trajectories (with their flow arrows).

dxdt=-8y,dydt=18x

Short Answer

Expert verified

The solution is9x2+4y2=c.

Step by step solution

01

Find phase plane equation

Here the system is;

dxdt=-8ydydt=18x

And the phase plane equation is;

dydx=-9x4y

02

Solve the equation

Here the equation is.

dydx=-9x4y

Solving by variable separating. Then,

9x2+4y2=c

Since, dxdt=-8yand x increases when y<0 and x decreases when y>0.

03

Sketch some trajectories

This is the required result.

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Most popular questions from this chapter

In Problems 29 and 30, determine the range of values (if any) of the parameter that will ensure all solutions x(t), and y(t) of the given system remain bounded as t→+∞.

dxdt=λx-y,dydt=3x+y

Falling Object. The motion of an object moving vertically through the air is governed by the equation\(\frac{{{{\bf{d}}^{\bf{2}}}{\bf{y}}}}{{{\bf{d}}{{\bf{t}}^{\bf{2}}}}}{\bf{ = - g - }}\frac{{\bf{g}}}{{{{\bf{V}}^{\bf{2}}}}}\frac{{{\bf{dy}}}}{{{\bf{dt}}}}\left| {\frac{{{\bf{dy}}}}{{{\bf{dt}}}}} \right|\)where y is the upward vertical displacement and V is a constant called the terminal speed. Take \({\bf{g = 32ft/se}}{{\bf{c}}^{\bf{2}}}\)and V = 50 ft/sec. Sketch trajectories in the yv-phase plane for \( - 100 \le {\bf{y}} \le 100, - 100 \le {\bf{v}} \le 100\)starting from y = 0 and y = -75, -50, -25, 0, 25, 50, and 75 ft/sec. Interpret the trajectories physically; why is V called the terminal speed?

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 11–14, solve the related phase plane differential equation for the given system. Then sketch by hand several representative trajectories (with their flow arrows).

dxdt=3ydydt=2x

In Problems 3–6, find the critical point set for the given system.

dxdt=x2-2xy,dydt=3xy-y2
See all solutions

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