Let y = y(x) be the solution of the differential equation (x^2 - x x^2 - 1 )dy + (y(x - x^2 - 1 ) - x)dx = 0 , x 1 . If y(1) = 1 , then the greatest integer less than y( 5 ) is _______.

Step-by-step solution

The given differential equation is: (x^2 - x x^2 - 1 )dy + (y(x - x^2 - 1 ) - x)dx = 0 Dividing the entire equation by dx and rearranging, we get: x(x - x^2 - 1 ) dy dx + y(x - x^2 - 1 ) = x Dividing by x(x - x^2 - 1 ) , we obtain a linear differential equation: dy dx + 1 x y = 1 x - x^2 - 1 The integrating factor (I.F.) is: I.F. = e^ 1 x dx = e^ x = x Multiplying the differential equation by the integrating factor x , we get: x dy dx + y = x x - x^2 - 1 d dx (xy) = x(x + x^2 - 1 ) (x - x^2 - 1 )(x + x^2 - 1 ) d dx