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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: Q5E (page 271)

In Problems 3–6, find the critical point set for the given system.
$\frac{d x}{d t} = x^{2} - 2 x y, \frac{d y}{d t} = 3 x y - y^{2}$

Short Answer

Expert verified

The only critical point is (0,0).

Step by step solution

01

Find critical points

Consider the system as;

$\frac{d x}{d t} = x^{2} - 2 x y \frac{d y}{d t} = 3 x y - y^{2}$

For finding the critical points put the system equal to 0.

So,

$x^{2} - 2 x y = 0 3 x y - y^{2} = 0$

02

Solve for x and y

Put the value of y in another equation and solve, then;

$\begin{array}{rcl} x (x - 2 y) & = & 0 \\ x & = & 0 o r 2 y \\ y (3 x - y) & = & 0 \\ y & = & 0 o r 3 x \end{array}$

Thus, the only critical points are (0,0).

This is the required result.

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Most popular questions from this chapter

Let $A = D - 1, B = D + 2, C = D^{2} + D - 2,$ where $D = \frac{d}{dt}$ . For $y = t^{3} - 8$ , compute

(a) $A [y]$

(b) $B [A [y]]$

(c) $B [y]$

(d) $A [B [y]]$

(e) $C [y]$

Let A, B and C represent three linear differential operators with constant coefficients; for example,

$A : = a_{2} D^{2} + a_{1} D + a_{0}, B : = b_{2} D^{2} + b_{1} D + b_{0}, C : = c_{2} D^{2} + c_{1} D + c_{0},$

Where the a’s, b’s, and c’s are constants. Verify the following properties:

(a) Commutative laws:

$A + B = B + A, A B = B A$

(b)Associative laws:

$(A + B) + C = A + (B + C), (AB) C = A (BC) .$

(c)Distributive law: $A (B + C) = AB + AC$

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).

$y'' (t) + y (t) - y (t)^{3} = 0$

In Problems 1 and 2, verify that the pair x(t), and y(t) is a solution to the given system. Sketch the trajectory of the given solution in the phase plane.

$\frac{dx}{dt} = 3 y^{3}, \frac{dy}{dt} = y; x (t) = e^{3 t}, y (t) = e^{t}$

In Section 3.6, we discussed the improved Euler’s method for approximating the solution to a first-order equation. Extend this method to normal systems and give the recursive formulas for solving the initial value problem.

See all solutions

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