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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: Q4E (page 271)

In Problems 3–6, find the critical point set for the given system.
$\frac{d x}{d t} = y - 1 \frac{d y}{d t} = x + y + 5$

Short Answer

Expert verified

The only critical points are (-6,1).

Step by step solution

01

Find critical points

Consider the system as;

$\frac{d x}{d t} = y - 1 \frac{d y}{d t} = x + y + 5$

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

So,

$\begin{array}{rcl} y - 1 & = & 0 \\ y & = & 1 \\ x + y + 5 & = & 0 \\ x + y & = & - 5 \end{array}$

02

Solve for x and y

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

$\begin{array}{rcl} x + 1 & = & - 5 \\ x & = & - 6 \end{array}$

Thus, the only critical point is (-6, 1).

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Most popular questions from this chapter

In Problem 31, assume that no solution flows out of the system from tank B, only 1 L/min flows from A into B, and only 4 L/min of brine flows into the system at tank A, other data being the same. Determine the mass of salt in each tank at the time $t ⩾ 0$ .

In Problems 1–7, convert the given initial value problem into an initial value problem for a system in normal form.

$y^{6} (t) = {[y' (t)]}^{3} - \sin (y (t)) + e^{2 t}; y (0) = y' (0) = . . . . . . = y^{(5)} (0) = 0$

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) $C [y]$

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

$x' = yz; x (0) = 0, y' = - xz; y (0) = 1, z' = \frac{- xy}{2}; z (0) = 1,$

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).

$\frac{d x}{d t} = - 8 y, \frac{d y}{d t} = 18 x$

See all solutions

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