 Good morning friends, I am Purva and today I will help you with the following question form the differential equation of the family of Parabolas having vertex at origin and axis along positive y-axis Let us now begin with the solution So here we are required to find the equation of the family of parabolas having vertex at the origin and axis along positive y-axis So first let us find the equation of one of the member of this family Let p denote the family of such parabolas and let 0 comma a Be the focus of the member of the family So here in this figure we have p is the family of such parabolas and 0 comma a is the focus of the member of the family Then its equation is x square is equal to 4 a y where a is an arbitrary Constant and we mark this as equation one Since there is only one constant so we shall differentiate equation one only once to get the required differential equation so differentiating one With respect to x we get Differentiating x square we get 2x is Equal to differentiating 4 a y we get 4 a into dy by dx And this implies Counseling of 2 here we get 2 so x is equal to 2 a into Now dy by dx is equal to y dash so we get y dash and this implies x upon 2 y dash is equal to a now substituting this value of a in equation one We get x square is equal to 4 into x upon 2 y dash into y Canceling out x from both the sides we get x here and Canceling out 2 in denominator and numerator we get 2 in numerator So we get this implies x is equal to 2 y upon y dash and this implies x into y dash is equal to 2 y Which further implies x into y dash minus 2 y is equal to 0 which is the required differential equation Hence we get our answer as x into y dash Minus 2 y is equal to 0. This is our answer. Hope you have understood the solution I and take care