Let y = y(x) be the solution of the differential equation dy dx = 2x ^4 x + ^4 x satisfying y ( 2 ) = 4 . Let f(x) = |y(x)| . If the sum of all the points x (0, 10 ) where the function f(x) is not differentiable is K , then K is equal to ____.

Step-by-step solution

First, simplify the denominator of the differential equation: ^4 x + ^4 x = ( ^2 x + ^2 x)^2 - 2 ^2 x ^2 x = 1 - 1 2 ^2 2x = 1 - 1 2 (1 - ^2 2x) = 1 + ^2 2x 2 The differential equation becomes: dy dx = 2 2x 1 + ^2 2x Integrating both sides: dy = 2 2x 1 + ^2 2x , dx Let 2x = t , then -2 2x , dx = dt : y = -dt 1 + t^2 = - ⁻¹(t) + C y(x) = - ⁻¹( 2x) + C Using the initial condition y ( 2 ) = 4 : 4 = - ⁻¹( ) + C 4 = - ⁻¹(-1) + C 4 = 4 + C C = 0 Thus, y(x) = - ⁻¹( 2x) . The function f(x) = |y(x)| = | ⁻¹( 2x)| . The funct