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# Section 2.5 Answers

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1. $$2x^{3}y^{2}=c$$

2. $$3y\sin x+2x^{2}e^{x}+3y=c$$

3. Not exact

4. $$x^{2}-2xy^{2}+4y^{3}=c$$

5. $$x+y=c$$

6. Not exact

7. $$2y^{2}\cos x+3xy^{3}-x^{2}=c$$

8. Not exact

9. $$x^{3}+x^{2}y+4xy^{2}+9y^{2}=c$$

10. Not exact

11. $$\ln |xy|+x^{2}+y^{2}=c$$

12. Not exact

13. $$x^{2}+y^{2}=c$$

14. $$x^{2}y^{2}e^{x}+2y+3x^{2}=c$$

15. $$x^{3}e^{x^{2}+y}-4y^{3}+2x^{2}=c$$

16. $$x^{4}e^{xy}+3xy=c$$

17. $$x^{3}\cos xy+4y^{2}+2x^{2}=c$$

18. $$y=\frac{x+\sqrt{2x^{2}+3x-1}}{x^{2}}$$

19. $$y=\sin x-\sqrt{1-\frac{\tan x}{2}}$$

20. $$y=\left(\frac{e^{x}-1}{e^{x}+1} \right)^{1/3}$$

21. $$y=1+2\tan x$$

22. $$y=\frac{x^{2}-x+6}{(x+2)(x-3)}$$

23. $$\frac{7x^{2}}{2}+4xy+\frac{3y^{2}}{2}=c$$

24. $$(x^{4}y^{2}+1)e^{x}+y^{2}=c$$

29.

1. $$M(x,y)=2xy+f(x)$$
2. $$M(x,y)=2(\sin x+x\cos x)(y\sin y +\cos y)+f(x)$$
3. $$M(x,y) = ye^{x}-e^{y}\cos x+f(x)$$

30.

1. $$N(x,y)=\frac{x^{4}y}{2}+x^{2}+6xy+g(y)$$
2. $$N(x,y)=\frac{x}{y}+2y\sin x+g(y)$$
3. $$N(x,y)=x(\sin y+y\cos y)+g(y)$$

33. $$B=C$$

34. $$B=2D,\quad E=2C$$

37.

1. $$2x^{2}+x^{4}y^{4}+y^{2}=c$$
2. $$x^{3}+3xy^{2}=c$$
3. $$x^{3}+y^{2}+2xy=c$$

38. $$y=-1-\frac{1}{x^{2}}$$

39. $$y=x^{3}\left(\frac{-3(x^{2}+1)+\sqrt{9x^{4}+34x^{2}+21}}{2} \right)$$

40. $$y=-e^{-x^{2}}\left(\frac{2x+\sqrt{9-5x^{2}}}{3} \right)$$

44.

1. $$G(x,y)=2xy+c$$
2. $$G(x,y)=e^{x}\sin y+c$$
3. $$G(x,y)=3x^{2}y-y^{3}+c$$
4. $$G(x,y)=-\sin x\sinh y+c$$
5. $$G(x,y)=\cos x\sinh y+c$$