Question

Does does d(xy)/xy mean in regards to differential equations?

Answer 1

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

Answer 2

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

Answer 3

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

Answer 4

hang on a sec - need to do something

Answer 5

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

Answer 6

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

Answer 7

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

Answer 8

You can integrate this now.

Answer 9

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

Answer 10

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

Answer 11

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

Answer 12

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

Answer 13

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

Answer 14

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

Answer 15

Why do you say the derivative of xy is 1?

Answer 16

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

Answer 17

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

Answer 18

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

Answer 19

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.

Answer 20

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

Answer 21

Okay, if I said,

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

Answer 22

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

Answer 23

whatever we are integrating with repect to

Answer 24

or im not sure lol

Answer 25

lol, who's got a headache?

Answer 26

i just got here and i do

Answer 27

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

Answer 28

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

Answer 29

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

Answer 30

This is what we're doing here.

Answer 31

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

Answer 32

Here the 'something' is (xy).

Answer 33

oh yes i get the something thing now

Answer 34

good!

Answer 35

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

Answer 36

Yes.

Answer 37

But even better, understand how we get that form.

Answer 38

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

Answer 39

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.

Answer 40

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

Answer 41

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

Answer 42

You're welcome.

Answer 43

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.

Answer 44

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.

Answer 45

if I could become your fan again lokisan, i would

Answer 46

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.

Answer 47

hey, thanks myininaya ;)

Answer 48

Can I hire you over skype? lol

Answer 49

lol, just log on here. I lurk around.

Answer 50

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

Answer 51

But

Answer 52

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

Answer 53

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

Answer 54

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

Answer 55

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.

Answer 56

Cheers nadeem :)

Answer 57

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

Answer 58

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

Answer 59

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 ;)

Answer 60

thank you mstud.

Answer 61

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

Answer 62

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

Answer 63

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

Answer 64

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

Answer 65

I would integrade which would be x

Answer 66

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

Answer 67

No problem

Answer 68

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.

Answer 69

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

Answer 70

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

Answer 71

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

Answer 72

Yeah, you can rearrange what I gave you.

Answer 73

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.

Answer 74

Yes, you'll need a different method.

Answer 75

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

Answer 76

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?

Answer 77

Yes

Answer 78

That's right.

Answer 79

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

Answer 80

Just like what I did above.

Answer 81

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

Answer 82

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

Answer 83

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

Answer 84

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

Answer 85

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

Answer 86

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

Answer 87

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

Answer 88

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

Answer 89

If you sleep on it, it will sink in.

Answer 90

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

Answer 91

Okay...good luck!