Why Is the Solution to dy/dx = -x/y Expressed as y^2 + x^2 = c?

[fran1942]

Hello, regarding the differential equation: "dy/dx = -x/y"
The general solution is y^2+x^2 = c.

I am wondering why it is expressed this way instead of "y=-x^2+c" ?
I thought you had to separate the x and y to opposite sides of the equation ?

Thanks for any help.

 

[tiny-tim]

hello fran1942!

(you mean y = √(c - x²) )

… I thought you had to separate the x and y to opposite sides of the equation ?

no, there's nothing special about y …

the answer is a curve (in this case, a circle), and x² + y² = c is a more natural way of describing a curve

 

[JJacquelin]

Hello, regarding the differential equation: "dy/dx = -x/y"
The general solution is y^2+x^2 = c.

I am wondering why it is expressed this way instead of "y=-x^2+c" ?
I thought you had to separate the x and y to opposite sides of the equation ?

Thanks for any help.

dy/dx = -x/y
y*dy = -x*dx
y²/2= -x²/2 +C
y² = -x² +c (c=2C)
y²+x² = c

 

[HallsofIvy]

Hello, regarding the differential equation: "dy/dx = -x/y"
The general solution is y^2+x^2 = c.

I am wondering why it is expressed this way instead of "y=-x^2+c" ?

Well, it wouldn't be expressed that way because those are not at all the same!
I presume you meant y= \sqrt{c- x^2}. The difficulty with that is that it is only "half" of the solution- the other half would be y= -\sqrt{c- x^2}.

I thought you had to separate the x and y to opposite sides of the equation ?

Thanks for any help.

No, differential equations often have solutions that are not functions.