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Fundamentals Of Differential Equations And Boundary Value Problems

R. Kent Nagle, Edward B. Saff, Arthur David Snider

$Math Studyset Vaia Explanations$ Math

9th Edition · 616 pages

ISBN: 9780321977069

Chapter 5: Q28E (page 272)

Figure 5.16 displays some trajectories for the system $\frac{d x}{d t} = y, \frac{d y}{d t} = - x + x^{2}$ What types of critical points (compare Figure 5.12 on page 267) occur at (0, 0) and (1, 0)?

Short Answer

Expert verified

The points are (0,0) and saddle point(1,0).

Step by step solution

Find the critical point.

Here the equation is:

$\frac{d x}{d t} = y \frac{d y}{d t} = - x + x^{2}$

And

$\frac{d y}{d x} = \frac{- x + x^{2}}{y}$

Here the points are (0,0) and saddle point (1,0).

Sketch Directional field.

This is the required result.

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Most popular questions from this chapter

In Problems 25 – 28, use the elimination method to find a general solution for the given system of three equations in the three unknown functions x(t), y(t), and z(t).

$x' = x + 2 y + z, y' = 6 x - y, z' = - x - 2 y - z$

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).

$\frac{dx}{dt} = (y - x) (y - 1) \frac{dy}{dt} = (x - y) (x - 1)$

Referring to the coupled mass-spring system discussed in Example , suppose an external force $E (t) = 37 \cos 3 t$ is applied to the second object of mass $1 kg$ . The displacement functions $x (t)$ and $y (t)$ now satisfy the system

$(16) (2 x'' (t) + 6 x (t) - 2 y (t) = 0, (17) (y'' (t) + 2 y (t) - 2 x (t) = 37 \cos 3 t$

(a) Show that $x (t)$ satisfies the equation $(18) x^{(4)} (t) + 5 x'' (t) + 4 x (t) = 37 \cos 3 t$

(b) Find a general solution $x (t)$ to the equation (18). [Hint: Use undetermined coefficients with $x_{p} = Acos 3 t + Bsin 3 t$ .]

(d) If both masses are displaced2mto the right of their equilibrium positions and then released, find the displacement functions $x (t)$ and $y (t)$ .

In Problems 3 – 18, use the elimination method to find a general solution for the given linear system, where differentiation is with respect to t.

$2 x' + y' - x - y = e^{- t}, x' + y' + 2 x + y = e^{t}$

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

$\frac{dx}{dt} = x - y, \frac{dy}{dt} = x^{2} + y^{2} - 1$

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Figure 5.16 displays some trajectories for the system dxdt=y,dydt=-x+x2What types of critical points (compare Figure 5.12 on page 267) occur at (0, 0) and (1, 0)?

Short Answer

Step by step solution

Find the critical point.

Sketch Directional field.

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Most popular questions from this chapter

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Figure 5.16 displays some trajectories for the system $\frac{d x}{d t} = y, \frac{d y}{d t} = - x + x^{2}$ What types of critical points (compare Figure 5.12 on page 267) occur at (0, 0) and (1, 0)?