14.22: Sección 2.3 Respuestas

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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)\(x_{0}y_{0}>0\text{ and }x_{0}y_{0}\neq 1\)

todos\((x_{0},y_{0})\)
\((x_{0},y_{0})\)woth\(y_{0}\neq 0\)

6. b) todos\((x_{0}, y_{0})\)

7. b) todos\((x_{0}, y_{0})\)

8. b) de\((x_{0}, y_{0})\) tal manera que\(x_{0}\neq 4y_{0}\)

todos\((x_{0}, y_{0})\)
todos\((x_{0}, y_{0})\neq (0,0)\)

10.

todos\((x_{0}, y_{0})\)
todo\((x_{0}, y_{0})\) con\(y_{0}\neq\pm 1\)

11. b) todos\((x_{0}, y_{0})\)

12. b) todos\((x_{0}, y_{0})\) tales que\((x_{0}, y_{0}) >0\)

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

16. \(y=\left(\frac{3}{5}x+1 \right)^{5/3},\quad -\infty <x<\infty\), es una solución.

También,\[y=\left\{\begin{array}{cc}{0,}&{-\infty <x\leq -\frac{5}{3}}\\{(\frac{3}{5}x+1)^{5/3},}&{-\frac{5}{3}<x<\infty }\end{array} \right.\nonumber\] es una solución. Para cada uno\(a\geq \frac{5}{3}\), la siguiente función es también una solución:\[y=\left\{\begin{array}{cc}{(\frac{3}{5}(x+a))^{5/3},}&{-\infty <x<-a,}\\{0,}&{-a\leq x\leq -\frac{5}{3}}\\{(\frac{3}{5}x+1)^{5/3},}&{-\frac{5}{3}<x<\infty }\end{array} \right.\nonumber\]

17.

todos\((x_{0}, y_{0})\)
todo\((x_{0}, y_{0})\) con\(y_{0}\neq 1\)

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

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

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

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

20.

La solución es única en\((0,\infty )\). Está dado por
\[y=\left\{\begin{array}{cc}{1,}&{0<x\leq \sqrt{5}}\\{1-(x^{2}-5)^{3/2},}&{\sqrt{5}<x<\infty }\end{array} \right.\nonumber\]
\[y=\left\{\begin{array}{cc}{1,}&{-\infty <x\leq\sqrt{5},}\\{1-(x^{2}-5)^{3/2},}&{\sqrt{5}<x<\infty }\end{array} \right.\nonumber\]es una solución de (A) on\((-\infty ,\infty )\). Si\(\alpha\geq 0\), entonces\[y=\left\{\begin{array}{cc}{1+(x^{2}-\alpha ^{2})^{3/2},}&{-\infty <x<-\alpha ,}\\{1,}&{-\alpha\leq x\leq\sqrt{5},}\\{1-(x^{2}-5)^{3/2},}&{\sqrt{5}<x<\infty ,}\end{array}\right.\nonumber\] y también\[y=\left\{\begin{array}{cc}{1-(x^{2}-\alpha ^{2})^{3/2},}&{-\infty <x<-\alpha ,}\\{1,}&{-\alpha\leq x\leq\sqrt{5},}\\{1-(x^{2}-5)^{3/2},}&{\sqrt{5}<x<\infty ,}\end{array} \right.\nonumber\] son soluciones de (A) en\((-\infty ,\infty)\).