Let y = y(x) be the solution of the differential equation x ( y x )dy = (y ( y x ) - x )dx , y(1) = 2 and let = ( y(e¹²) e¹² ) . Then the number of integral values of p , for which the equation x^2 + y^2 - 2px + 2py + + 2 = 0 represents a circle of radius r 6 , is __________.

Step-by-step solution

Given the differential equation: x ( y x )dy = (y ( y x ) - x )dx Rearranging the terms, we get: x ( y x )dy - y ( y x )dx = -xdx Dividing both sides by x^2 : ( y x ) ( xdy - ydx x^2 ) = - dx x ( y x )d ( y x ) = - dx x Integrating both sides: - ( y x ) = - |x| - C ( y x ) = |x| + C Using the initial condition y(1) = 2 : ( 2 ) = (1) + C C = 0 Thus, the solution to the differential equation is: ( y x ) = |x| We need to find = ( y(e¹²) e¹² ) . Substituting x = e¹² : = (e¹²) = 12 The given equation of the circle is: x