A.4.5: Section 4.5 Answers
- Page ID
- 43764
1. \(y'=\frac{2xy}{x^{2}+3y^{2}}\)
2. \(y'=-\frac{y^{2}}{(xy-1)}\)
3. \(y'=-\frac{y(x^{2}+y^{2}-2x^{2}\ln |xy|)}{x(x^{2}+y^{2}-2y^{2}\ln |xy|)}\)
4. \(xy'-y=-\frac{x^{1/2}}{2}\)
5. \(y'+2xy=4xe^{x^{2}}\)
6. \(xy'+y=4x^{3}\)
7. \(y' − y = \cos x − \sin x\)
8. \((1+x^{2})y'-2xy=(1-x)^{2}e^{x}\)
10. \(y'g-yg'=f'g-fg'\)
11. \((x-x_{0})y'=y-y_{0}\)
12. \(y'(y^{2}-x^{2}+1)+2xy=0\)
13. \(2x(y-1)y'-y^{2}+x^{2}+2y=0\)
14.
- \(y=-81+18x,\: (9,81)\quad y=-1+2x,\: (1,1)\)
- \(y = −121 + 22x,\: (11, 121)\quad y = −1 + 2x,\: (1, 1)\)
- \(y = −100 − 20x,\: (−10, 100)\quad y = −4 − 4x,\: (−2, 4)\)
- \(y = −25 − 10x,\: (−5, 25)\quad y = −1 − 2x,\: (−1, 1)\)
15. (e) \(y=\frac{5+3x}{4},\:(-3/5,4/5)\quad y=-\frac{5-4x}{3},\: (4/5,-3/5)\)
17.
- \(y=-\frac{1}{2}(1+x),\: (1,-1);\quad y=\frac{5}{2}+\frac{x}{10},\: (25,5)\)
- \(y=\frac{1}{4}(4+x),\: (4,2)\quad y=-\frac{1}{4}(4+x),\: (4,-2)\)
- \(y = \frac{1}{2}(1 + x),\: (1, 1)\quad y = \frac{7}{2} + \frac{x}{14},\: (49, 7)\)
- \(y = −\frac{1}{2}(1 + x),\: (1, −1)\quad y = −\frac{5}{2} −\frac{x}{10},\: (25, −5)\)
18. \(y=2x^{2}\)
19. \(y=\frac{cx}{\sqrt{|x^{2}=1|}}\)
20. \(y=y_{1}+c(x-x_{1})\)
21. \(y=-\frac{x^{3}}{2}-\frac{x}{2}\)
22. \(y=-x\ln |x|+cx\)
23. \(y=\sqrt{2x+4}\)
24. \(y=\sqrt{x^{2}-3}\)
25. \(y=kx^{2}\)
26. \((y-x)^{3}(y+x)=k\)
27. \(y^{2}=-x+k\)
28. \(y^{2}=-\frac{1}{2}\ln (1+2x^{2})+k\)
29. \(y^{2}=-2x-\ln (x-1)^{2}+k\)
30. \(y=1+\sqrt{\frac{9-x^{2}}{2}};\text{ those with }c>0\)
33. \(\tan ^{-1}\frac{y}{x}-\frac{1}{2}\ln (x^{2}+y^{2})=k\)
34. \(\frac{1}{2}\ln (x^{2}+y^{2})+(\tan\alpha )\tan ^{-1}\frac{y}{x}=k\)