Verifying Relations: Solving x^2 + y^2 = 1 | Homework Help

[bengaltiger14]

Homework Statement

Verify that the relation x^2 + y^2 = 1 is a solution to the differential equation:
dy/dx = xy/(x^2 - 1)

Can anyone point me in the right direction on how to begin to solve this problem? Do I take integral of the DE and just plug into equation?

 

[rock.freak667]

You could take x^2 + y^2 = 1 and find dy/dx and see if it is the same as the DE

 

[bengaltiger14]

dy/dx is just 2x+2y. So it does not equal correct?

 

[rock.freak667]

dy/dx is just 2x+2y. So it does not equal correct?

2x+2y(dy/dx)=0 is what you get when you differentiate implicitly with repect to x. Now just find dy/dx and do a bit of algebra.

 

[bengaltiger14]

I'm sorry but I'm really slow at DE's.

2x+2y(dy/dx)=0

What do you mean find dy/dx. I thought dy/dx was 2x+2y=0.

Are you saying to solve for dy/dx in the first equation?

 

[rock.freak667]

Are you saying to solve for dy/dx in the first equation?

Yes, make dy/dx the subject of the equation.

EDIT: is the relation supposed to be y²-x²=1 ?

 

[bengaltiger14]

2y(dy/dx) = -2x

divide by 2y:

dy/dx = -2x/2y

 

[rock.freak667]

2y(dy/dx) = -2x

divide by 2y:

dy/dx = -2x/2y

So dy/dx= -x/y right?

Now multiply the numerator and denominator by y.

(Forget my edit in the previous post)

 

[bengaltiger14]

dy/dx = -xy

That is the answer to the problem? So the DE is not a solution.

 

[rock.freak667]

dy/dx = -xy

That is the answer to the problem? So the DE is not a solution.

If dy/dx=-x/y and you multiply by y/y, do you really get dy/dy=-xy or -xy/y²?

 

[bengaltiger14]

yeah, your right... dy/dx = -xy/y^2

But the top y would just cancel and the y^2 would become y again so why do that?

 

[rock.freak667]

yeah, your right... dy/dx = -xy/y^2

But the top y would just cancel and the y^2 would become y again so why do that?

Don't cancel out anything. Keep it the way it is. Now from the relation x²+y²=1, what is y² equal to?

 

[bengaltiger14]

y^2 = 1-x^2

 

[rock.freak667]

y^2 = 1-x^2

Now in the equation

$$\frac{dy}{dx}=\frac{-xy}{y^2}$$


Replace y².

 

[bengaltiger14]

dy/dx = -xy/1-x^2

So they do equal. If you multiply by -1, it gives the DE.

 

[rock.freak667]

dy/dx = -xy/1-x^2

So they do equal. If you multiply by -1, it gives the DE.

and if it gives the DE, it is a solution of it.

 

[bengaltiger14]

Cool..Thank you very much for your help and patience.