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Question
Form the differential equation representing the family of parabolas having vertex at origin and axis along positive direction of x-axis.
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Solution
The equation of the parabola having vertex at origin and axis along the positive direction of x-axis is given by
y2 =4ax .....(1)
Since there is only one parameter, so we differentiate it only once.
Differentiating with respect to x, we get
\[2y\frac{dy}{dx} = 4a\]
Substituting the value of 4a in (1), we get
\[y^2 = 2y\frac{dy}{dx} \times x\]
\[ \Rightarrow y^2 = 2xy\frac{dy}{dx}\]
\[ \Rightarrow y^2 - 2xy\frac{dy}{dx} = 0\]
\[\]
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