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

• • William F. Trench
• Andrew G. Cowles Distinguished Professor Emeritus (Mathematics) at Trinity University
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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. Figure 1.3E.1 : A direction field for $$y'= {\frac{x}{y}}$$ 2. Figure 1.3E.2 : A direction field for $${y'= {2xy^2\over1+x^2}}$$ 3. Figure 1.3E.3 : A direction field for $$y'=x^2(1+y^2)$$ 4. Figure 1.3E.4 : A direction field for $$y'= {1\over1+x^2+y^2}$$ 5. Figure 1.3E.5 : A direction field for $$y'=-(2xy^2+y^3)$$ 6. Figure 1.3E.6 : A direction field for $$y'=(x^2+y^2)^{1/2}$$ 7. Figure 1.3E.7 : A direction field for $$y'=\sin xy$$ 8. Figure 1.3E.8 : A direction field for $$y'=e^{xy}$$ 9. Figure 1.3E.9 : A direction field for $$y'=(x-y^2)(x^2-y)$$ 10. Figure 1.3E.10 : A direction field for $$y'=x^3y^2+xy^3$$ 11. Figure 1.3E.11 : A direction field for $$y'=\sin(x-2y)$$

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

12. $$y'=y(y-1); \quad \{-1\le x\le 2,\ -2\le y\le2\}$$

13. $$y'=2-3xy; \quad \{-1\le x\le 4,\ -4\le y\le4\}$$

14. $$y'=xy(y-1); \quad \{-2\le x\le2,\ -4\le y\le 4\}$$

15. $$y'=3x+y; \quad \{-2\le x\le2,\ 0\le y\le 4\}$$

16. $$y'=y-x^3; \quad \{-2\le x\le2,\ -2\le y\le 2\}$$

17. $$y'=1-x^2-y^2; \quad \{-2\le x\le2,\ -2\le y\le 2\}$$

18. $$y'=x(y^2-1); \quad \{-3\le x\le3,\ -3\le y\le 2\}$$

19. $$y'= {x\over y(y^2-1)}; \quad \{-2\le x\le2,\ -2\le y\le 2\}$$

20. $$y'= {xy^2\over y-1}; \quad \{-2\le x\le2,\ -1\le y\le 4\}$$

21. $$y'= {x(y^2-1)\over y}; \quad \{-1\le x\le1,\ -2\le y\le 2\}$$

22. $$y'=- {x^2+y^2\over1-x^2-y^2}; \quad \{-2\le x\le2,\ -2\le y\le 2\}$$

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

$T' = -k(T-T_m) \tag{A}$

and

$G'=-\lambda G+r\tag{B}$

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

$y'=- ay+b, \tag{C}$

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 $$b=kT_m$$, and (B) is of the form (C) with $$y=G$$, $$a=\lambda$$, 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 $\{0\le t\le T,\ c\le y\le d\}$

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

$P'=aP(1-\alpha P) \tag{A}$

and

$I'=rI(S-I) \tag{B}$

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

$y'=ay-by^2, \tag{C}$

where $$a$$ and $$b$$ are positive constants. Thus, (A) is of the form (C) with $$y=P$$, $$a=a$$, and $$b=a\alpha$$, 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 $\{0\le t\le T,\ 0\le y\le d\}$

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 $\{0\le t\le T,\ c\le y\le 0\}$

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.