Question

The differential equation of the family of curves cy² = 2x + c, where c is an arbitrary constant is

(A) y(dy/dx) = 1

(B) (dy/dx)² + y(d²y/dx²) = 0

(C) y² = 2xy(dy/dx) + 1

(D) None of these

Answer 1

Answer is (C) y² = 2xy(dy/dx) + 1

Differentiating the given differential equation

cy² = 2x + c (1)

c. 2y . dy/dx = 2

⇒ c = 1/y. dx/dy

Putting this value of c in Eq. (1), we get

1/y . dx/dy . y² = 2x + 1/y . dx/dy

y² = 2xy(dy/dx) + 1