If y = y (x) satisfies ( 2+ x 1+y ) dy d x =- x . such that y(0)=2 , then y ( 2 ) is equal to

Step-by-step solution

The differential equation ( 2+ x 1+y ) dy d x =- x is separable. Separating variables yields dy 1+y = - x 2+ x d x . Integrating both sides, 1 1+y dy = - x 2+ x d x . The left side integrates to |1+y| , while the right side, using substitution u = 2+ x , gives - |2+ x| . Combining the constants, |1+y| = - |2+ x| + C . Exponentiating both sides and letting A = e^C , the general solution becomes (1+y)(2+ x) = A . Using the initial condition y(0)=2 , substitution gives (1+2)(2+ 0) = A , so A = 6 . The particular solut