Solve the initial value problem: `x(dy)/(dx)+y=xcosx+sinx ,\ y\ (pi/2)=1`

Explanation

`quad x frac{d y}{d x}+y=x cos x+sin x `


`text { or } frac{d y}{d x}+frac{1}{x} cdot y=cos x+frac{1}{x} sin x`


`therefore quad text { I.F. }=e^{int frac{1}{x} d x}=e^{log x}=x`


`therefore text { Solution is } x y=int(x cos x+sin x) d x`


`text { or } quad x y=x sin x+C`


`therefore quad y=sin x+C frac{1}{x}`


`therefore=frac{pi}{2}, y=1 text { or } 1=1+C frac{2}{pi} text { or } C=0`


`therefore text { Solution is } y=sin x`