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Chapter 1: Q2 E (page 1)

The direction field for dydx=2x+yas shown in figure 1.13.

  1. Sketch the solution curve that passes through (0, -2). From this sketch, write the equation for the solution.

b. Sketch the solution curve that passes through (-1, 3).

c. What can you say about the solution in part (b) as x鈫�+鈭�? How about x鈫�-鈭�?

Short Answer

Expert verified
  1. The graph is drawn below, and the equation is y=-2-2x.
  2. The graph is drawn below, and the equation isy=1-2x.
  3. The solutions become infinite whenx鈫�+鈭�orx鈫�-鈭�.

Step by step solution

01

Step 1(a): Find the curve by point (0,-2)

Givendydx=2x+y鈥�......(1)

Put the value of the point (0,-2) in equation (1)

m=2(0)-2=-2

The curve is(y+2)=-2(x-0)

Hence the solution is y=-2-2x.

By putting the different values of x, get the values of y.

02

Step 2(b): Find the curve by point (-1,3).

Givendydx=2x+y.鈥�.....(2)

Put the value of the point (-1,3) in equation (2)

m=2(-1)-3=1

The curve isy=1-2x

Hence the solution is y=1-2x.

03

Step 3(c): Discuss the solution in part (b) as x→+∞ and x→-∞

As x鈫�鈭�the solution becomes infinite. And whenx鈫�-鈭�the solution also becomes infinite and has an asymptotey=-2-2x.

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Most popular questions from this chapter

The motion of a set of particles moving along the x鈥慳xis is governed by the differential equation dxdt=t3-x3, where xt denotes the position at time t of the particle.

猞� If a particle is located at x=1 when t=1 , what is its velocity at this time?

猞� Show that the acceleration of a particle is given by d2xdt2=3t2-3t3x2+3x5.

猞� If a particle is located at x=2when t=2.5, can it reach the location x=1at any later time?

[Hint: t3-x3=(t-x)(t2+xt+x2).]

(a) For the initial value problem (12) of Example 9. Show that 蠒1(x)=0 and蠒2(x)=(x-2)3are solutions. Hence, this initial value problem has multiple solutions. (See also Project G in Chapter 2.)

(b) Does the initial value problemy'=3y23,y(0)=10-7have a unique solution in a neighbourhood ofx=0?

(a) Show that y2+x-3=0 is an implicit solution todydx=-12y on the interval (-鈭�,3).

(b) Show thatxy3-xy3sinx=1 is an implicit solution todydx=xcosx+sinx-1y3x-xsinx on the interval (0,蟺2).

Mixing.Suppose a brine containing 0.2 kg of salt per liter runs into a tank initially filled with 500 L of water containing 5 kg of salt. The brine enters the tank at a rate of 5 L/min. The mixture, kept uniform by stirring, is flowing out at the rate of 5 L/min (see Figure 2.6).

(a)Find the concentration, in kilograms per liter, of salt in the tank after 10 min. [Hint:LetAdenote the number of kilograms of salt in the tank attminutes after the process begins and use the fact that

rate of increase inA=rate of input- rate of exit.

A further discussion of mixing problems is given in Section 3.2.]

(b)After 10 min, a leak develops in the tank and an additional liter per minute of mixture flows out of the tank (see Figure 2.7). What will be the concentration, in kilograms per liter, of salt in the tank 20 min after the leak develops? [Hint:Use the method discussed in Problems 31 and 32.]

The logistic equation for the population (in thousands) of a certain species is given by dpdt=3p-2p2.

猞� Sketch the direction field by using either a computer software package or the method of isoclines.

猞� If the initial population is 3000 [that is, p(0) = 3], what can you say about the limiting population?

猞� If p(0)=0.8, what is limt鈫�+鈭�p(t)?

猞� Can a population of 2000 ever decline to 800?
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