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

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

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Equation of parabola having vertex at origin and axis along positive y-axis is
x^2= 4ay                        (i)
Now, differentiate w.r.t. c
2x= 4a\frac{dy}{dx}\\ \\ \frac{dy}{dx} =y^{'}= \frac{x}{2a}
a=\frac{x}{2y^{'}}                           -(ii)
Put value from equation (ii) in (i)
x^2= 4y.\frac{x}{2y^{'}}\\ xy^{'}-2y = 0
Therefore, the required equation is xy^{'}-2y = 0

Posted by

Gautam harsolia

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