## anonymous one year ago If dy/dx = sin^2( piy/4) and y = 1 when x = 0, then find the value of x when y = 3.

1. freckles

do you know how to solve the given differential equation?

2. anonymous

im not too sure how to manipulate dy/dx

3. anonymous

i know i have to isolate the variables and integrate and solve for y right?

4. freckles

$\frac{dy}{dx}=\sin^2(\frac{\pi}{4} y) \\ \frac{dy}{\sin^2(\frac{\pi}{4} y)}=dx \\$ like do you know how to ingrate both sides of this equation?

5. anonymous

well the right side is just x but im not to sure on integrating the left side

6. freckles

do you know the derivative of cot(x)?

7. anonymous

-csc^2(x)?

8. freckles

$\frac{d}{dy} \cot(u)=-\csc^2(u) \cdot \frac{du}{dy} \\ \text{ where } u=u(y)$

9. freckles

$\int\limits_{}^{} \csc^2(\frac{\pi}{4} y) dy \\ u=\frac{\pi}{4} y \\ du=\frac{\pi}{4} dy \\ \int\limits \csc^2(u) \frac{4}{\pi} du$

10. freckles

do you see how to continue

11. anonymous

why did you inverse the pi/4

12. freckles

because if du=pi/4 dy then dy=4/pi du

13. freckles

i just multiplied 4/pi on both sides to isolate dy

14. anonymous

okay okay hold up slow down , why are we integrating csc^2(piy/4)

15. freckles

because that was the left hand side of your equation once you separated the variables

16. anonymous

okay okay i see, give me a second to get my thoughts together

17. freckles

$\frac{1}{\sin(x)}=\csc(x)$

18. anonymous

yea but the sine was squared wouldn't that affect that

19. freckles

so you know that equation I just mentioned holds don't you think the equation will still hold if you square both sides

20. freckles

$(\frac{1}{\sin(x)})^2=\csc^2(x) \\ \frac{1}{\sin^2(x)}=\csc^2(x)$

21. anonymous

okay sorry i was thinking the square would be negative because it was in the denominator

22. anonymous

i see what you mean though

23. anonymous

okay so we are integrating $\csc^2(\frac{ \pi y }{ 4 })$

24. anonymous

oh you were doing substitution , i see

25. anonymous

okay so back to $\int\limits\limits_{}^{} \csc^2(\frac{\pi}{4} y) dy \\ u=\frac{\pi}{4} y \\ du=\frac{\pi}{4} dy \\ \int\limits\limits \csc^2(u) \frac{4}{\pi} du$ should be do another substitution?

26. freckles

on what? that is just -4/pi*cot(u)+C where u=pi/4y

27. anonymous

$\frac{ 4 }{ \pi } \int\limits (-\ln(\csc(x) + \cot(x)) + C)^2$

28. anonymous

is that right?

29. anonymous

minus the integral sign

30. freckles

where did you get that answer

31. freckles

remember you yourself said the d/dx cot(x) =-csc^2(x)

32. freckles

or you could say -d/dx cot(x)=csc^2(x)

33. anonymous

34. anonymous

are these wrong then?

35. freckles

$\int\limits\limits\limits_{}^{} \csc^2(\frac{\pi}{4} y) dy \\ u=\frac{\pi}{4} y \\ du=\frac{\pi}{4} dy \\ \int\limits\limits\limits \csc^2(u) \frac{4}{\pi} du \\ \frac{-4}{\pi} \cot(u)+C \\ \frac{-4}{\pi} \cot(\frac{\pi}{4}y)+C$

36. freckles

nope those all look great

37. anonymous

arghh i feel stupid

38. freckles

anyways we also have to integrate the other side

39. freckles

$\frac{-4}{\pi} \cot(\frac{\pi}{4} y)+C=x$

40. anonymous

-_- i was just typing that, okay so now just isolate y right

41. freckles

your second job is to find C for the point (x=0,y=1)

42. freckles

i would not solve for y i think that would cause more work then needed

43. freckles

besides you want to find x

44. freckles

for when y=3

45. freckles

which is the third part of the problem third and final

46. anonymous

$C = \pi$

47. anonymous

- pi

48. freckles

$\frac{-4}{\pi} \cot(\frac{\pi}{4} y)+C=x \\ (0,1)=(x,y) \\ \frac{-4}{\pi}\cot(\frac{\pi}{4}(1))+C=0 \\ \frac{-4}{\pi} \cot(\frac{\pi}{4})+C=0 \\ \frac{-4}{\pi}(1)+C=0 \\ \frac{-4}{\pi}+C=0$

49. anonymous

:(

50. freckles

it's okay... no frowns here!

51. anonymous

alright well c = 4

52. freckles

$\frac{-4}{\pi} \cot(\frac{\pi}{4} y)+C=x \\ \frac{-4 }{\pi} \cot(\frac{\pi}{4} y)+\frac{4}{\pi}=x$ last step find x for when y is 3

53. freckles

well no

54. freckles

c=4/pi

55. anonymous

okay let me just take a breath , my frustration is getting the better of me here

56. anonymous

okay so now we just plug c into the solution and plug y = 3 and find x right

57. freckles

right

58. freckles

i will let you do that and I will come back in check in like 5 minutes or less

59. anonymous

okay im getting 8/pi

60. freckles

that sounds right

61. freckles

4/pi+4/pi 2*4/pi 8/pi

62. anonymous

okay great , thanks for the help sorry for being such a dummy :P

63. freckles

i'm going to go for tonight after this problem sleep time but before i go do you have any last questions on this problem?

64. freckles

you aren't a dummy probably just frustrated and trying to learn a new subject

65. anonymous

no im goo thanks again !

66. freckles

the one part is hard enough being frustrated makes it tad harder

67. freckles

anyways peace and i gave you a medal to award you for your effort

68. anonymous

:) night