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Question

(A) The general equation of a parabola whose axis is parallel to the $y$-axis is given by $y = Ax^2 + Bx + C$,where $A, B, C$ are arbitrary constants.

Vedclass · Accepted Answer

$\frac{d^3 y}{d x^3} = 0$

Vedclass · Answer

(A) The general equation of a parabola whose axis is parallel to the $y$-axis is given by $y = Ax^2 + Bx + C$,where $A, B, C$ are arbitrary constants.To find the differential equation,we differentiate with respect to $x$ three times.First derivative: $\frac{dy}{dx} = 2Ax + B$.Second derivative: $\frac{d^2y}{dx^2} = 2A$.Third derivative: $\frac{d^3y}{dx^3} = 0$.Since there are $3$ arbitrary constants,the order of the differential equation is $3$. Thus,the equation is $\frac{d^3y}{dx^3} = 0$.

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

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