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

`(dy)/(dx)\left ( (x^2+y^2)/(2* y)-y \right )=x`

Explanation:

Given that

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

Now by differentiating the above equation with respect to x

`2* x+2* y* (dy)/(dx)=2* c* (dy)/(dx)`

`x+ y* (dy)/(dx)= c* (dy)/(dx)-----(2)`

Form the above equation (1)

`c=(x^2+y^2)/(2* y)`

Putting the value of c in the equation (2)

`x+ y* (dy)/(dx)= (x^2+y^2)/(2* y)* (dy)/(dx)\\(dy)/(dx)\left ( (x^2+y^2)/(2* y)-y \right )=x`

Thus the differential equation of the given curve will be

`(dy)/(dx)\left ( (x^2+y^2)/(2* 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.