Question

Determine the differential equation of the family of curves x² +y² = 2cy.
Use geobra to draw the family of curves for c = -3,-2,-1,2,3​

Answer 1

Answer:

$\dfrac{dy}{dx}\left ( \dfrac{x^2+y^2}{2\times y}-y \right )=x$

Step-by-step explanation:

Given that

$x^2+y^2=2cy----(1)$

Now by differentiating the above equation with respect to x

$2\times x+2\times y\times \dfrac{dy}{dx}=2\times c\times \dfrac{dy}{dx}$

$x+ y\times \dfrac{dy}{dx}= c\times \dfrac{dy}{dx}-----(2)$

Form the above equation (1)

$c=\dfrac{x^2+y^2}{2\times y}$

Putting the value of c in the equation (2)

$x+ y\times \dfrac{dy}{dx}= \dfrac{x^2+y^2}{2\times y}\times \dfrac{dy}{dx}\\\dfrac{dy}{dx}\left ( \dfrac{x^2+y^2}{2\times y}-y \right )=x$

Thus the differential equation of the given curve will be

$\dfrac{dy}{dx}\left ( \dfrac{x^2+y^2}{2\times y}-y \right )=x$

Curve for different value of c :

x² +y² = 2 c y

The above equation is an equation of a circle.That circle is having center on the y-axis.