 Good morning friends and Pooja and today we will discuss the following question from the differential equation of the family of ellipses, I will foci on y-axis and center at origin Let's begin with the solution now Now as earlier we shall find the equation of one member of the family of ellipses Having foci on y-axis and center at the origin So let 0 comma ae and 0 comma minus ae be the foci Let 0 comma 0 be the center Then equation of such an ellipse will be x square by v square plus y square by a square is equal to 1 Here 2a and 2b are major and minor axis and so a and b are constants And let us mark this equation x square by v square plus y square by a square is equal to 1 as equation 1 Since there are two arbitrary constants a and b So in order to eliminate them we shall differentiate equation 1 two times So differentiating 1 with respect to x we get differentiating x square by v square we get 2x upon v square plus differentiating y square by a square we get 2y into dy by dx upon a square is equal to differentiate in 1 we get 0 Or we can write this as 2y upon a square into now dy by dx is equal to y dash so we have y dash is equal to minus 2x upon b square Canceling out two from both the sides we get this implies y into y dash upon x is equal to minus a square upon b square and we mark this as equation 2 Now differentiating equation 2 with respect to x we get now on left hand side we apply quotient and product rule So we have denominator that is x into differentiation of numerator now in numerator we apply product rule So we get first that is y into differentiation of second that is differentiation of y dash which gives y double dash plus second that is y dash into differentiation of first that is differentiation of y gives y dash minus numerator that is y into y dash into differentiation of denominator that is 1 upon denominator square that is x square and this is equal to now differentiating minus a square by v square gives 0 This implies now opening the bracket we get x into y into y double dash plus x into now y dash into y dash gives y dash square minus y into y dash into 1 gives y into y dash and this is equal to now x square into 0 gives 0 As this equation is free from constants a and b therefore this is the required differential equation Hence the required answer is x into y into y double dash plus x into y dash square minus y into y dash is equal to 0 this is our answer Hope you have understood the solution. Bye and take care