Let y=y(x) be the solution of the differential equation 2 x ~d y ~d x = 2 x-4 y x, x (0, 2 ) . If y ( 3 )=0 , then y^ ( 4 )+y ( 4 ) is equal to ________.

Step-by-step solution

aligned & d y d x +2 y x= x & I.F. =e^ 2 x d x = ^2 x & y ^2 x= x ^2 x d x & = x x d x & = x+C & C=-2 aligned aligned & y= x-2 ^2 x & y ( 4 )= 1 2 -1 & y^ =- x+4 x x & y^ ( 4 )=- 1 2 +2 & y^ ( 4 )+y ( 4 )=1 aligned