Question

Find the differential equation of the family of parabolas with vertex at (0, –1) and having axis along the y-axis

Answer 1

Equation of the family of parabolas with vertex at (0, –1) and having axis along the y-axis is

(x – 0)² = 4a(y + 1)

x² = 4a(y + 1)  .......(1)

x² = 4 ay + 4a

Differentiating equation (1) with respect to ‘x’, we get
2x = 4a y’

`(2x)/(y"'")` = 4a

Substituting 4a value in equation (1), we get

x² = `(2x)/(y"'") (y + 1)`

`x^2/x = 2/(y"'") (y + 1)`

x = `2/(y"'") (y + 1)`

xy’ = 2(y + 1)

xy’ = 2y + 2

xy’ – 2y – 2 = 0 is a required differential equation.