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Question
Form the differential equation of the family of parabolas having vertex at origin and axis along positive y-axis.
Solution
The equation of the parabola having vertex at origin and axis along the positive direction of y-axis is given by
x2 =4ay .....(1)
Since there is only one parameter, so we differentiate it only once.
Differentiating with respect to x, we get
\[2x = 4ay'\]
\[ \Rightarrow 4a = \frac{2x}{y'}\]
Substituting the value of 4a in (1), we get
\[x^2 = \frac{2x}{y'} \times y\]
\[ \Rightarrow xy' = 2y\]
\[ \Rightarrow xy' - 2y = 0\]
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