If `y+d/(dx)(x y)=x(sinx+logx)`, find `y(x)`.

Explanation

The given differential equation is
`y+d/(dx)(xy) = x(sinx+logx)`
i.e., `x(dy)/(dx)+2/xy=sinx+logx`.............(1)
This is a linear different equation
I.F.`=e^(2int1/xdx)= e^(2logx)=x^(2)`............(2)
Thus, solution is given by
`yx^(2)=intx^(2)(sinx+logx)dx+c`
`=-x^(2)cosx+2xsinx+2cosx+x^(3)/(3)logx-x^(3)/9+c`
or `y=-cosx+2/xsinx+2/x^(2)cosx+x/3logx-x/3+c/x^(2)`