Let y = y(x) be the solution of the differential equation dy dx + y x = 2x , with y(0) = 1 . Then the maximum value of y(x) in the interval [0, 3 ] is equal to

Step-by-step solution

The given differential equation is a linear differential equation of the form dy dx + P(x)y = Q(x) , where P(x) = x and Q(x) = 2x . Integrating factor (IF) = e^ x dx = e^ | x| = x . The general solution is given by: y x = 2x x dx + C y x = 2 x x x dx + C y x = 2 x dx + C y x = -2 x + C Given y(0) = 1 , we have: 1 (0) = -2 (0) + C 1 = -2 + C C = 3 Thus, the particular solution is: y x = 3 - 2 x y(x) = 3 x - 2 ^2 x To find the maximum value of y(x) on [0, 3 ] , we differentiate y(x) with respect to x : y'(x) = -3 x +