Find the differential equation representing the family of curves y = aebx + 5, where a and b are arbitrary constants.

Given the curve, y = aebx + 5
or, y = (ae5)ebx
or, y = cebx [ Putting ae5 = c]. (1)
Now differentiating both sides with respect to x we get,
dy/dx = cbex.
or, b = (1/y)(dy/dx) (2). [ Using (1)]
Now again differentiating both sides with respect to x we get,
d2y/dx2 = cb2ebx
or, d2y/dx2 = y × (1/y2) × (dy/dx)2 [ Using (1) and (2)]
or, d2y/dx2 = (1/y) × (dy/dx)2