Question

The equation which represents the system of parabolas whose axis is parallel to y-axis satisfies the differential equation

• A. \(\frac{d^3y}{dx^3} = 0\)
• B. \(\frac{d^3y}{dx^3} + \frac{d^2y}{dx^2} = x+y\)
• C. \(\frac{d^2y}{dx^2} + xy = 4ax\)
• D. \(\frac{dy}{dx} + xy = x^2\)

Hint:

The order of the differential equation representing a family of curves is equal to the number of independent arbitrary constants in the general equation of the family.

Answer 1

The Correct Option is A

Step 1: General Equation of Parabola:
The general equation of a parabola with its axis parallel to the y-axis is: \[ y = Ax^2 + Bx + C \] Here, \(A, B, C\) are three arbitrary constants.
Step 2: Eliminate Constants:
To find the differential equation, we need to differentiate the equation as many times as there are arbitrary constants (3 times). 1st Derivative: \[ \frac{dy}{dx} = 2Ax + B \] 2nd Derivative: \[ \frac{d^2y}{dx^2} = 2A \] 3rd Derivative: \[ \frac{d^3y}{dx^3} = 0 \] The resulting differential equation is \(\frac{d^3y}{dx^3} = 0\).