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.
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In Exercises 12 - 22 construct a direction field and plot some integral curves in the indicated rectangular region.
12. y′=y(y−1);{−1≤x≤2, −2≤y≤2}
13. y′=2−3xy;{−1≤x≤4, −4≤y≤4}
14. y′=xy(y−1);{−2≤x≤2, −4≤y≤4}
15. y′=3x+y;{−2≤x≤2, 0≤y≤4}
16. y′=y−x3;{−2≤x≤2, −2≤y≤2}
17. y′=1−x2−y2;{−2≤x≤2, −2≤y≤2}
18. y′=x(y2−1);{−3≤x≤3, −3≤y≤2}
19. y′=xy(y2−1);{−2≤x≤2, −2≤y≤2}
20. y′=xy2y−1;{−2≤x≤2, −1≤y≤4}
21. y′=x(y2−1)y;{−1≤x≤1, −2≤y≤2}
22. y′=−x2+y21−x2−y2;{−2≤x≤2, −2≤y≤2}
23. By suitably renaming the constants and dependent variables in the equations
T′=−k(T−Tm)
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 {0≤t≤T, c≤y≤d}
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(1−αP)
and
I′=rI(S−I)
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′=ay−by2,
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 {0≤t≤T, 0≤y≤d}
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 {0≤t≤T, c≤y≤0}
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.