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

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A)
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• Equation of a parabola having vertex at origin at axis along the positive axis is $x^2 = 4ay$
Step 1:
Given: $x^2 = 4ay$-------(1)
Differentiating on both sides we get,
$2x = 4ay'$
$x = 2ay'$
$a =\large\frac{ x}{2y'}$
Step 2:
Substituting in equ(1) we get
$x^2 = 4. [\large\frac{x}{2y'}]$$.y$
On simplifying and rearranging we get
$xy' -2y = 0$
This is the required equation