Solve: `(dy)/(dx)+y/x=cos x+(sinx)/x`

Explanation

`{I} . {F}=e^{int p d x}=e^{sin x}`


The required solution is `y` (I.F) `=int {Q}(` I.F `) d x+c` `y e^{sin x}=int sin x cos x e^{sin x} d x+c`


Put `sin x=t`. Then `cos x d x=d t`


`therefore y e^{sin x}=int t e^{t} d t+c`


`=t e^{t}-e^{t}+c` (Using integration by parts)


`Rightarrow y e^{sin x}=sin x e^{sin x}-e^{sin x}+c`


(or) `y e^{sin x}=e^{sin x}(sin x-1)+c`