The differential equation represents:

⇒ y dy = (c – x)dx

Integrate

⇒ y dy = (c – x)dx

⇒ ∫y dy = ∫(c – x)dx

⇒ ∫y dy = ∫c dx – ∫x dx

k is the integration constant

⇒ x² + y² = 2cx + k

⇒ x² + y² – 2cx – k = 0

This is equation of circle because there is no ‘xy’ term and the coefficient of x² and y² is same.

This equation represents family of circles because for different values of c and k we will get different circles