Single Choice

The differential equation of all parabolas whose axes are parallel to $$\mathrm{y}$$ -axis is:

A$$\displaystyle \dfrac{d^{3}y}{dx^{3}}=0$$
Correct Answer
B$${\dfrac{d^{2}x}{dy^{2}}}=0$$
C$$\displaystyle \dfrac{d^{3}y}{dx^{3}}+\frac{d^{2}y}{dx^{2}}=0$$
D$$\displaystyle \dfrac{d^{2}y}{dx^{2}}+2\frac{dy}{dx}=0$$

Solution

Equation of parabola is $$ (x-h)^{2} = 4a (y-k)$$

Differentiating w.r.t. x, we get,

$$\Rightarrow 2(x-h) = 4ay_{1} $$

$$\Rightarrow (x-h) = 2ay_{1}$$

$$\Rightarrow 1 =2ay_{2}$$

$$\Rightarrow y_{2} = \dfrac{1}{2a}$$

Differentiating w.r.t. x, we get,

$$\Rightarrow y_{3}=0$$

$$\Rightarrow\dfrac{d^{3}y}{dx^{3}} = 0$$

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