Answer to Question #59070 in Differential Equations for ABJ

Question

3 Derive the differential equation associated with the primitive
y=Ax^3+Bx^2 + Cx + D
where A, B, C and D are arbitrary constants.

(a) D^3y/dx^2= 0
(b) d^4y/dx^4 + d^3y/dx^3 = 0
( c) d^3y/dx^3 + d^2y/dx^2 = 0
(d) d^4y/dx^4 = 0
5 Derive the differential equation for the area bounded by the arc of a curve, the x- axis, and the two ordinates, one fixed and one variable, is equal to trice the length of the arc between the ordinates
(I) y=2√4+ (dx/dy)^2
(II) Y =√1+(d^2y/dx^2)^2
(III) Y =2 √1+ (dy/dx)^2
(IV) y=3 √2+(dy/dx)^2
6 Find the differential equation of all straight lines at a unit distance from the origin
(i) (x dy/dx − y)^2=1/2)^2
(II) (x dy/dx − y)^2 =1+ (dy/dx)^2
(III) (3x dy/dx − y)^2=3+(dy/dx)^2
(IV) (2x d^2y/dx^2 − y)^2=1+(dy/dx)^2