## anonymous 5 years ago Does does d(xy)/xy mean in regards to differential equations?

1. anonymous

That just means the differential with respect to (xy), like du/u.

2. anonymous

So how would I solve that on paper? Would I take the derivative of xy and then take the integral of the entire thing?

3. anonymous

The whole problem is d(xy)/xy+dy=0

4. anonymous

hang on a sec - need to do something

5. anonymous

You can make it something more palatable for you by expanding the differential. That might be best. So, you'd write,

6. anonymous

$\frac{d(xy)}{xy}=\frac{ydx+xdy}{xy}=\frac{dx}{x}+\frac{dy}{y}$

7. anonymous

The equation is $\frac{dx}{x}+\frac{dy}{y}+dy=0 \rightarrow \frac{dx}{x}=-(\frac{1}{y}+1)dy$

8. anonymous

You can integrate this now.

9. anonymous

$\ln x = -(\ln y +y)+c \rightarrow x=e^{-\ln y - y-c}=\frac{1}{y}e^{-y}e^{-c}=\frac{Ce^{-y}}{y}$

10. anonymous

+c in the exponentiation, but it doesn't matter since it just becomes the constant at the end.

11. anonymous

You didn't do any integrating or differentation on that last part right? Just moved things around algebraically?

12. anonymous

Note, this could have been obtained in a couple of steps from the first method I mentioned, namely,$\frac{d(xy)}{xy}+dy=0 \rightarrow \ln xy +y = C \rightarrow xy.e^y=C \rightarrow x=\frac{C}{y}e^{-y}$

13. anonymous

Which last part? Once the 'dx' and 'dy' differentials disappear, there's no more integration.

14. anonymous

Nevermind. I think I see. I basically get the derivative of xy, which is 1. Then integrate 1/xy which is ln|xy|

15. anonymous

Why do you say the derivative of xy is 1?

16. anonymous

Hmm .. doesn't one normally want the answer to a diff eq on the form y=... ? I see you wrote x=...

17. anonymous

If you saw $\int\limits_{}{}\frac{du}{u}$you'd know what to do. You'd recognize it as ln u.

18. anonymous

Well, this would be transcendental in y, which is why it wasn't solved explicitly for y.

19. anonymous

Re. the du/u integral, the d(xy)/(xy) thing is in the SAME FORM...so you do to it what you would do in the situation du/u.

20. anonymous

I'm just having trouble figuring out with d(xy)/xy means. Is that just something I memorize du/u=ln u?

21. anonymous

Okay, if I said, "The anti-derivative of the differential of *something*, divided by that *something* is..?" what would you say to me?

22. anonymous

the integral of f(x) / f(x) ?

23. myininaya

whatever we are integrating with repect to

24. myininaya

or im not sure lol

25. anonymous

26. myininaya

i just got here and i do

27. anonymous

Me. I've been struggling with this for over 10 hours now. I hate online school.

28. anonymous

In mathematics, it's all about *forms*.

29. anonymous

We always try to reduce more complicated problems into forms we already have solutions to.

30. anonymous

This is what we're doing here.

31. anonymous

We know that, whenever you have the form, $\int\limits_{}{}\frac{d(something)}{(something)}$the solution is $\ln (something) + c$

32. anonymous

Here the 'something' is (xy).

33. myininaya

oh yes i get the something thing now

34. anonymous

good!

35. anonymous

ok , so its just a form I need to memorize basically

36. anonymous

Yes.

37. anonymous

But even better, understand how we get that form.

38. anonymous

But, look, if it's a pain for you and you stress in an exam, just expand the differential like I did above.

39. anonymous

Unfortunetly thats just the tip of the iceberg in my lack of understanding. I'm in an 8 week online Calculus II class and struggling very badly.

40. anonymous

But thank you for the patience and the help. I think I understand a little bit better now.

41. anonymous

There's heaps of online resources. This site for one. Paul's Online Maths Notes are good, as well as www.khanacademy.org.

42. anonymous

You're welcome.

43. anonymous

I've watched all those videos and looked. They help somewhat, but when I work specific problems I guess recognizing what method to use is where I struggle.

44. anonymous

Yes, that's called 'the problem of fluency' in mathematics. You need to do two things: 1) understand the mathematics and 2) recognize what you're being shown so you can access what you know. Sounds like number 2 is hassling you. That's good, though, because it can be fixed.

45. myininaya

if I could become your fan again lokisan, i would

46. anonymous

Haha I think I'm also having difficulty in the understanding at some aspect as well. The bad part is I have little time to become fluent and understand before I move onto the next subject. I would also fan you 10 times.

47. anonymous

hey, thanks myininaya ;)

48. anonymous

Can I hire you over skype? lol

49. anonymous

lol, just log on here. I lurk around.

50. anonymous

The fluency part is 'easily' fixable because all it requires is doing problems.

51. anonymous

But

52. anonymous

there are two types of problem: (1) closed and (2) open.

53. anonymous

Well I'm doing them. For instance right now I'm trying to apply either seperation of variables or integrating combinations to this problem. I think I know what to do but I'm stuck

54. anonymous

good stuff lokisan....... you couldn't have said it any better!

55. anonymous

Closed problems are the ones you're probably used to, like, if the sides of a rectangle are 7 and 3 units, what's the perimeter? You can work that out easily since it's plug and play. But if someone says, "The perimeter of a rectangle is 20cm. What are it's dimensions?" people come unstuck.

56. anonymous

57. anonymous

Hmm ok well in that same topic then... what if you didn't have d(something/(something) and it was just d(something)

58. anonymous

For instance d(xy)=-x/11 dx

59. anonymous

Well, Lokisan got one more fan for solving this differential eq :) I hadn't fanned him before (and is till now the only user here I am a fan of ;)

60. anonymous

thank you mstud.

61. anonymous

re. your question scotty, what would you do if you saw,$\int\limits_{}{}dx$?

62. anonymous

My current problem is 11xdy+11ydx+xdx=0

63. anonymous

I know, I'm trying to make it easy on you.

64. anonymous

I can move the xdx over, factor the 11 and then group xdy +ydx to be d(xy)

65. anonymous

I would integrade which would be x

66. anonymous

Just give me a sec. to sort something non-mathematical out.

67. anonymous

No problem

68. anonymous

Okay, back. The reason I asked you about $\int\limits_{}{} dx$is because I wanted you to tie this idea of 'forms' from the last situation to this one. You've made life hard for yourself by going the route you did.

69. anonymous

$\int\limits_{}{}d(xy)=\int\limits_{}{}-\frac{x}{11}dx \rightarrow xy=-\frac{x^2}{22}+c \rightarrow y=-\frac{x}{22}+\frac{c}{x}$

70. anonymous

There's that 'form' thing going on again in d(xy).

71. anonymous

I think the route I went was the completely wrong way because that isn't the answer lol. According to my book its 22xy+x^2=C

72. anonymous

Yeah, you can rearrange what I gave you.

73. anonymous

11xdy+11ydx+xdx=0 My first instinct is to want to move the xdx to the other side, but I already know this isn't seperable. So I need to use another method.

74. anonymous

Yes, you'll need a different method.

75. anonymous

It's a lot messier than just recognizing the form of what you've been given and just exploiting it.

76. anonymous

Ok so I think I can combine differentials to integrate them as a unit right? the 11xdy+11ydx can combine to form 11d(xy) right?

77. anonymous

Yes

78. anonymous

That's right.

79. anonymous

Ok so from this point, can I just integrate each term?

80. anonymous

Just like what I did above.

81. anonymous

Ok so you're probably going to kill me, but d(xy) basically means the differential of our "function"?

82. anonymous

or do I just leave it alone? d(xy) just becomes xy..

83. anonymous

lol, d(xy) means the differential of xy.

84. anonymous

$\int\limits_{}{}d(xy)=(xy)+c$

85. anonymous

I think I'm getting hung up on the definition of a differential

86. anonymous

$\int\limits_{}{}dw=w+c$$\int\limits_{}{}d(\cos \theta)=\cos \theta + c$$\int\limits_{}{}d(sty)=sty+c$$\int\limits_{}{}d(gp^4.78-\frac{q}{a}+x^2e^{xy})=gp^4.78-\frac{q}{a}+x^2e^{xy}+c$

87. anonymous

I think that last thing you posted makes sense. I think I have it losesly in my mind what it means now

88. anonymous

See what I keep doing, no matter how complicated it gets?

89. anonymous

If you sleep on it, it will sink in.

90. anonymous

So it basically negates it. I got it. Thanks dude I'm going to try a few more examples

91. anonymous

Okay...good luck!