A continuous function y = f(x) defined on [0, 2 ] satisfies the differential equation x dy dx + 2y x = x x . Then the value of f ( 2 ) is equal to

Step-by-step solution

Given differential equation: x dy dx + 2y x = x x Dividing by x (for x 0 ), we get the standard linear form: dy dx + (2 x)y = x Integrating Factor (I.F.) = e^ 2 x dx = e^ 2 ( x) = ^2 x The general solution is: y ^2 x = x ^2 x dx + C Using the half-angle formula ^2 x = 1 - 2x 2 : x ^2 x dx = ( x 2 - x 2x 2 ) dx = x^2 4 - 1 2 ( x 2x 2 - 1 2x 2 dx ) = x^2 4 - x 2x 4 - 2x 8 So, the solution is: y ^2 x = x^2 4 - x 2x 4 - 2x 8 + C Since f(x) is continuous at x = 0 , y(0) is a finite value. Taking the limit as x 0 on both