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# A fun calculus question

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I'm back! Maybe..... Anywho, here's my problem:

So, I was messing around with slope fields and got this pretty cool field given by the following equation:

$$\frac{dy}{dx}=\frac{x+y}{x-y}$$

Now, naturally, I was interested in finding the original equation, but it gave me problems, mainly this:

$$\int_{}^{}xdy$$

I have no clue what tto do with this, seeing as all the integration techniques I know are useless. Help, please!

ThisGuy  May 31, 2015

#2
+26493
+10

I think it's more complicated than this Chris.  Your result can be written as x2 + y2 = constant.

This gives 2x + 2ydy/dx = 0  or dy/dx = -x/y.

Here's my attempt:

.

Alan  Jun 1, 2015
Sort:

#1
+82944
+5

Cross-multiply .......this gives........

(x - y) dy = (x + y) dx      integrate both sides

∫ ( x- y) dy  = ∫ (x + y)  dx

[On the left side, x is considered to be a constant......same with y on the right side]

xy  - y^2/2  =  x^2/2 + xy + C      subtract xy from each side

-y^2/2 =  x^2/2 + C1

(And C1 is just a constant )

y^2  = 2C1 - x^2   let 2C1  = C2

y = ±√[C2 - x^2)

I believe this might be correct......Alan....will you check it????

CPhill  May 31, 2015
#2
+26493
+10

I think it's more complicated than this Chris.  Your result can be written as x2 + y2 = constant.

This gives 2x + 2ydy/dx = 0  or dy/dx = -x/y.

Here's my attempt:

.

Alan  Jun 1, 2015
#3
+82944
0

OK...thanks, Alan....

CPhill  Jun 1, 2015
#4
+227
+5

Whoah... Okay, thanks... That was really involved, but I should've assumed it would be! Thanks for the help!

ThisGuy  Jun 1, 2015

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