Subjective Type

Find the differential equation of all the parabola having axis parallel to the $$x-$$axis.

Solution

Equation of parabola having axis parallel to x-axis is given by,

$$(y-\beta )^2=a(x-\alpha )$$

differentiating on both sides, we get,

$$2(y-\beta )=\dfrac{dy}{dx}=a$$

again differentiating on both sides, we get,

$$2(y-\beta )=\dfrac{d^2y}{dx^2}+2\left ( \dfrac{dy}{dx} \right )^2=0$$

eliminating $$\beta $$ from the above equations, we get,

$$a\dfrac{d^2y}{dx^2}+2\left ( \dfrac{dy}{dx} \right )^3=0$$

Is the required differential equation.

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