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1.3E: Direction Fields for First Order Equations (Exercises)

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Exercises for Section 1.3

In Exercises 1–11 a direction field is drawn for the given equation. Sketch some integral curves.

1.

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Figure 1.3E.1 : A direction field for y=xy

2.

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Figure 1.3E.2 : A direction field for y=2xy21+x2

3.

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Figure 1.3E.3 : A direction field for y=x2(1+y2)

4.

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Figure 1.3E.4 : A direction field for y=11+x2+y2

5.

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Figure 1.3E.5 : A direction field for y=(2xy2+y3)

6.

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Figure 1.3E.6 : A direction field for y=(x2+y2)1/2

7.

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Figure 1.3E.7 : A direction field for y=sinxy

8.

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Figure 1.3E.8 : A direction field for y=exy

9.

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Figure 1.3E.9 : A direction field for y=(xy2)(x2y)

10.

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Figure 1.3E.10 : A direction field for y=x3y2+xy3

11.

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Figure 1.3E.11 : A direction field for y=sin(x2y)

In Exercises 12 - 22 construct a direction field and plot some integral curves in the indicated rectangular region.

12. y=y(y1);{1x2, 2y2}

13. y=23xy;{1x4, 4y4}

14. y=xy(y1);{2x2, 4y4}

15. y=3x+y;{2x2, 0y4}

16. y=yx3;{2x2, 2y2}

17. y=1x2y2;{2x2, 2y2}

18. y=x(y21);{3x3, 3y2}

19. y=xy(y21);{2x2, 2y2}

20. y=xy2y1;{2x2, 1y4}

21. y=x(y21)y;{1x1, 2y2}

22. y=x2+y21x2y2;{2x2, 2y2}

23. By suitably renaming the constants and dependent variables in the equations

T=k(TTm)

and

G=λG+r

discussed in Section 1.2 in connection with Newton’s law of cooling and absorption of glucose in the body, we can write both as

y=ay+b,

where a is a positive constant and b is an arbitrary constant. Thus, (A) is of the form (C) with y=T, a=k, and b=kTm, and (B) is of the form (C) with y=G, a=λ, and b=r. We’ll encounter equations of the form (C) in many other applications in Chapter 2.

Choose a positive a and an arbitrary b. Construct a direction field and plot some integral curves for (C) in a rectangular region of the form {0tT, cyd}

of the ty-plane. Vary T, c, and d until you discover a common property of all the solutions of (C). Repeat this experiment with various choices of a and b until you can state this property precisely in terms of a and b.

24. By suitably renaming the constants and dependent variables in the equations

P=aP(1αP)

and

I=rI(SI)

discussed in Section 1.1 in connection with Verhulst’s population model and the spread of an epidemic, we can write both in the form

y=ayby2,

where a and b are positive constants. Thus, (A) is of the form (C) with y=P, a=a, and b=aα, and (B) is of the form (C) with y=I, a=rS, and b=r. In Chapter 2 we’ll encounter equations of the form (C) in other applications..

Choose positive numbers a and b. Construct a direction field and plot some integral curves for (C) in a rectangular region of the form {0tT, 0yd}

of the ty-plane. Vary T and d until you discover a common property of all solutions of (C) with y(0)>0. Repeat this experiment with various choices of a and b until you can state this property precisely in terms of a and b.

Choose positive numbers a and b. Construct a direction field and plot some integral curves for (C) in a rectangular region of the form {0tT, cy0}

of the ty-plane. Vary a, b, T and c until you discover a common property of all solutions of (C) with y(0)<0.

You can verify your results later by doing Exercise 2.2.27.


This page titled 1.3E: Direction Fields for First Order Equations (Exercises) is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by William F. Trench.

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