3.3E: Exercises

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Exercises

Exercise $\PageIndex{1}$

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

$clipboard_ecaa0db6f1fdef51a705e90b878c02022.png$
**Figure $\PageIndex{1}$**: A direction field for $y'= {\frac{x}{y}}$

$clipboard_e36fdf613776a431b1e8191cecfb75d19.png$
**Figure $\PageIndex{2}$**: A direction field for $ {y'= {2xy^2\over1+x^2}}$

$clipboard_e2e0f78fd7d0281deeb3df372a3dd82bf.png$
**Figure $\PageIndex{3}$**: A direction field for $y'=x^2(1+y^2)$

4.

$clipboard_eaeb583fc57656cdd45141b50283234ba.png$
**Figure $\PageIndex{4}$**: A direction field for $y'= {1\over1+x^2+y^2}$

5.

$clipboard_e773af5b2db36a2716990bc6547ea8350.png$
**Figure $\PageIndex{5}$**: A direction field for $y'=-(2xy^2+y^3)$

6.

$clipboard_e608ddd48ea3de96fabef8c8dbc9f2b95.png$
**Figure $\PageIndex{6}$**: A direction field for $y'=(x^2+y^2)^{1/2}$

7.

$clipboard_e4c24dceacf5c04358243ff0dbcfab51f.png$
**Figure $\PageIndex{7}$**: A direction field for $y'=\sin xy$

8.

$clipboard_ea448bb2833068eee5327e80c03f3bc30.png$
**Figure $\PageIndex{8}$**: A direction field for $y'=e^{xy}$

9.

$clipboard_e994c3c42adb6dd58d08b10ccb5b0549c.png$
**Figure $\PageIndex{9}$**: A direction field for $y'=(x-y^2)(x^2-y)$

10.

$clipboard_e06d564a692a46e7905532a83d418f113.png$
**Figure $\PageIndex{10}$**: A direction field for $y'=x^3y^2+xy^3$

11.

$clipboard_e1a0943243a0e768584defadf1c51a9d4.png$
**Figure $\PageIndex{11}$**: A direction field for $y'=\sin(x-2y)$

Answer: Add texts here. Do not delete this text first.

Exercise $\PageIndex{2}$

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

Answer: Add texts here. Do not delete this text first.

Exercise $\PageIndex{3}$

23. By suitably renaming the constants and dependent variables in the equations A: $T' = -k(T-T_m) $ and B: $G'=-\lambda G+r$ discussed in connection with Newton's

law of cooling and absorption of glucose in the body, we can write both as C: $y'=- ay+b$ 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 $b=kT_m$, 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.

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 $b$.

Answer: Add texts here. Do not delete this text first.

Exercise $\PageIndex{4}$

24. By suitably renaming the constants and dependent variables in the equations A: $P'=aP(1-\alpha P)$ and B: $I'=rI(S-I)$ discussed in connection with Verhulst's population model and the spread of an epidemic, we can write both in the form C: $y'=ay-by^2$ 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 $b=r$. Later, we'll encounter equations of the form (C) in other applications..

(a) 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 $b$.

(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 $y(0)<0$.

Answer: Add texts here. Do not delete this text first.

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