Let y = y(x) be the solution curve of the differential equation x dy dx + 2y x = 2x for x (0, 2 ) - 2 , 3 2 , satisfying the initial condition y ( 3 ) = 0 . If the sum of the abscissas of all the points of intersection of the curve y = y(x) with the line y = -1 is k , then the value of k is ____.

Step-by-step solution

The given differential equation is x dy dx + 2y x = 2x . Dividing by x , we get a linear differential equation: dy dx + (2 x)y = 2x x = 2 x The integrating factor (IF) is: IF = e^ 2 x , dx = e^ 2 | x| = ^2 x Multiplying the equation by the integrating factor and integrating: y ^2 x = 2 x ^2 x , dx y ^2 x = 2 x x , dx y ^2 x = 2 x + C Using the initial condition y ( 3 ) = 0 : 0 ^2 ( 3 ) = 2 ( 3 ) + C 0 = 2(2) + C C = -4 The solution curve is: y ^2 x = 2 x - 4 y = 2 x - 4 ^2 x To find the intersection with y = -1 , w