Question

Obtain the differential equation by eliminating arbitrary constants from the following equation.

y² = (x + c)³

Answer 1

y² = (x + c)³       ...(i)

Differentiating w.r.t. x, we get

`2y dy/dx = 3 (x+c)^2`        ...(ii)

Dividing (i) by (ii), we get

`y^2/(2y(dy/dx)) = ((x+c)^3)/(3(x+c)^2)`

∴ `y/(2(dy/dx)) = (x+c)/3`

∴ `x+c = (3y)/(2(dy/dx))`

∴ `c=-x +(3y)/(2(dy/dx))`

Substituting the value of c in (i), we get

`y^2 = [x+(-x+(3y)/(2(dy/dx)))]^3`

= `((3y)/(2(dy/dx)))^3`

∴ `y^2 = (27y^3)/(8(dy/dx)^3`

∴ `(dy/dx)^3 = (27y)/8`

∴ `dy/dx = 3/2root3y`