If y=y(x) is the solution of the differential equation x dy d x +2 y=x^2 satisfying y(1)=1 , then the value of y ( 1 2 ) is

Step-by-step solution

aligned & x ~d y ~d x +2 y=x^2 & ~d y ~d x + ( 2 x ) y=x & I.F. = e ^ 2 x ~d x = e ^ 2 x =x^2 aligned Solution of differential equation is aligned & y x^2= x x^2 ~d x+ c & y x^2= x^4 4 + c ...(i) & & y= x^2 4 + c x^2 ...(ii) aligned aligned & Given, y(1)=1 & 1= 1 4 + c & c = 3 4 aligned equation (i) becomes, aligned & y= x^2 4 + 3 4 x^2 & y ( 1 2 )= ( 1 2 )^2 4 + 3 4 ( 1 2 )^2 = 1 16 +3= 49 16 aligned