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.

 

[blue_raver22]

Hello, thank you for your question. The reason why the general solution is expressed as y^2+x^2 = c instead of y=-x^2+c is because this is a separable differential equation, meaning we can separate the variables and integrate each side separately. In this case, we can rewrite the equation as ydy = -xdx and then integrate both sides to get y^2/2 + x^2/2 = c. This is then simplified to y^2 + x^2 = c, as the constant c can be absorbed into the 2 in the denominator.

Separating the variables is a common method in solving differential equations, but it is not the only method. In some cases, it may be easier to solve the equation by keeping the variables on the same side. However, for this particular equation, separating the variables and integrating is the most straightforward approach. I hope this helps to clarify the reasoning behind the general solution. Let me know if you have any further questions.