A.2.3: Section 2.3 Answers

1. (b) $x_{0}\neq k\pi \;\;\;\;(k=\text{integer})$

2. (b) $(x_{0},y_{0})\neq (0,0)$

3. (b) $x_{0}y_{0}\neq (2k+1)\frac{\pi }{2} \;\;\;\; (k=\text{integer})$

4. (b) $x_{0}y_{0}>0\text{ and }x_{0}y_{0}\neq 1$

5.

1. all $(x_{0},y_{0})$
2. $(x_{0},y_{0})$ woth $y_{0}\neq 0$

6. (b) all $(x_{0}, y_{0})$

7. (b) all $(x_{0}, y_{0})$

8. (b) $(x_{0}, y_{0})$ such that $x_{0}\neq 4y_{0}$

9.

1. all $(x_{0}, y_{0})$
2. all $(x_{0}, y_{0})\neq (0,0)$

10.

1. all $(x_{0}, y_{0})$
2. all $(x_{0}, y_{0})$ with $y_{0}\neq\pm 1$

11. (b) all $(x_{0}, y_{0})$

12. (b) all $(x_{0}, y_{0})$ such that $(x_{0}, y_{0}) >0$

13. (b) all $(x_{0}, y_{0})$ with $x_{0}\neq 1,\quad y_{0}\neq (2k+1)\frac{\pi }{2}(k=\text{integer})$

16. $y=\left(\frac{3}{5}x+1 \right)^{5/3},\quad -\infty <x<\infty$, is a solution.

Also, $$y=\left\{\begin{array}{cc}{0,}&{-\infty <x\leq -\frac{5}{3}}\\[4pt]{(\frac{3}{5}x+1)^{5/3},}&{-\frac{5}{3}<x<\infty }\end{array} \right.\nonumber$$ is a solution. For every $a\geq \frac{5}{3}$, the following function is also a solution: $$y=\left\{\begin{array}{cc}{(\frac{3}{5}(x+a))^{5/3},}&{-\infty <x<-a,}\\[4pt]{0,}&{-a\leq x\leq -\frac{5}{3}}\\[4pt]{(\frac{3}{5}x+1)^{5/3},}&{-\frac{5}{3}<x<\infty }\end{array} \right.\nonumber$$

17.

1. all $(x_{0}, y_{0})$
2. all $(x_{0}, y_{0})$ with $y_{0}\neq 1$

18. $y_{1}=1; y_{2}=1+|x|^{3};y_{3}=1-|x|^{3};y_{4}=1+x^{3};y_{5}=1-x^{3}$

$$y_{6}=\left\{\begin{array}{cc}{1+x^{3},}&{x\geq 0,}\\[4pt]{1,}&{x<0}\end{array} \right.;\quad y_{7}=\left\{\begin{array}{cc}{1-x^{3},}&{x\geq 0,}\\[4pt]{1,}&{x<0}\end{array} \right.;\nonumber$$

$$y_{8}=\left\{\begin{array}{cc}{1,}&{x\geq 0,}\\[4pt]{1+x^{3},}&{x<0}\end{array} \right.;\quad y_{9}=\left\{\begin{array}{cc}{1,}&{x\geq 0,}\\[4pt]{1-x^{3},}&{x<0}\end{array} \right.\nonumber$$

19. $y=1+(x^{2}+4)^{3/2},\quad -\infty <x<\infty$

20.

1. The solution is unique on $(0,\infty )$. It is given by
   $$y=\left\{\begin{array}{cc}{1,}&{0<x\leq \sqrt{5}}\\[4pt]{1-(x^{2}-5)^{3/2},}&{\sqrt{5}<x<\infty }\end{array} \right.\nonumber$$
2. $$y=\left\{\begin{array}{cc}{1,}&{-\infty <x\leq\sqrt{5},}\\[4pt]{1-(x^{2}-5)^{3/2},}&{\sqrt{5}<x<\infty }\end{array} \right.\nonumber$$ is a solution of (A) on $(-\infty ,\infty )$. If $\alpha\geq 0$, then $$y=\left\{\begin{array}{cc}{1+(x^{2}-\alpha ^{2})^{3/2},}&{-\infty <x<-\alpha ,}\\[4pt]{1,}&{-\alpha\leq x\leq\sqrt{5},}\\[4pt]{1-(x^{2}-5)^{3/2},}&{\sqrt{5}<x<\infty ,}\end{array}\right.\nonumber$$ and $$y=\left\{\begin{array}{cc}{1-(x^{2}-\alpha ^{2})^{3/2},}&{-\infty <x<-\alpha ,}\\[4pt]{1,}&{-\alpha\leq x\leq\sqrt{5},}\\[4pt]{1-(x^{2}-5)^{3/2},}&{\sqrt{5}<x<\infty ,}\end{array} \right.\nonumber$$ are also solutions of (A) on $(-\infty ,\infty)$.