Question

Form the differential equation representing the family of curves given by where a is an arbitrary constant.

Answer 1

The differential equation is ( x−a ) 2 +2 y 2 = a 2 .

Simplify the above equation.

( x−a ) 2 +2 y 2 = a 2 x 2 + a 2 −2ax+2 y 2 = a 2 2 y 2 =2ax− x 2 (1)

Differentiate above equation with respect to x.

2y dy dx =2a−2x dy dx = a−x 2y dy dx = 2ax−2 x 2 4xy

Substitute the value of 2ax from equation (1).

dy dx = 2 y 2 + x 2 −2 x 2 4xy dy dx = 2 y 2 − x 2 4xy

Thus, the differential equation of family of curves is dy dx = 2 y 2 − x 2 4xy .