Form the differential equation of the family | vedclass

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Question

(A) The equation of the parabola having the vertex at the origin and the axis along the positive $y$-axis is:$x^{2}=4 a y$           $(1)$Differentiat

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$x y^{\prime}-2 y=0$

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(A) The equation of the parabola having the vertex at the origin and the axis along the positive $y$-axis is:$x^{2}=4 a y$           $(1)$Differentiating equation $(1)$ with respect to $x$,we get:$2 x=4 a y^{\prime}$           $(2)$Dividing equation $(2)$ by equation $(1)$,we get:$\frac{2 x}{x^{2}}=\frac{4 a y^{\prime}}{4 a y}$$\Rightarrow \frac{2}{x}=\frac{y^{\prime}}{y}$$\Rightarrow x y^{\prime}=2 y$$\Rightarrow x y^{\prime}-2 y=0$This is the required differential equation.

Form the differential equation of the family of parabolas having vertex at origin and axis along the positive $y$-axis.

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