If y = y (x) and ( 2+ x y +1 ) dy d x =- x , y(0)=1 , then y ( 2 )=

Step-by-step solution

Solution The differential equation is solved by separating variables: dy y+1 = - x 2+ x dx Integrating both sides: 1 y+1 dy = - x 2+ x dx The left integral is |y+1| . For the right, substitute u = 2+ x , du = x dx , yielding - |2+ x| . Combining constants of integration: |y+1| + |2+ x| = C Exponentiating gives (y+1)(2+ x) = K , with K constant. Applying y(0)=1 : (1+1)(2+0) = 4 = K . Thus, (y+1)(2+ x) = 4 . Evaluating at x = 2 , where 2 = 1 : (y( 2 )+1)(2+1) = 4 (y( 2 )+1) = 4 3 y( 2 ) = 1 3 . The answer is A .