Form the differential equation representing the family of curves y = a sin (x + b), where a, b are arbitrary constants.

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Given,

y = a sin (x + b) … (1)

Differentiating both sides of equation (1) with respect to x,

dy/dx = a cos (x + b) … (2)

Differentiating again on both sides with respect to x,

d2y/dx2= – a sin (x + b) … (3)

Eliminating a and b from equations (1), (2) and (3),

d2y/ dx2+ y = 0 … (4)

The above equation is free from the arbitrary constants a and b.

This the required differential equation.

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