The solution of the equation `(dy)/(dx)=(x(2logx+1))/(siny+ycosy)` is

Explanation

`frac{d y}{d x}=frac{x(2 log x+1)}{sin y+y cos y}`


`Rightarrow(sin y+y cos y) d y=x(2 log x+1) d x`


Integrate both sides


`int(sin y+y cos y) d y=int x(2 log x+1) d x`


`Rightarrow int sin y d y+int y cos y d y=2 int x log x d x+int x d x`


`Rightarrow-cos y+[y int cos y d y-int{frac{d}{d y}(y) int cos y d y} d y]=2[log x int x d x-int{frac{d}{d x}(log x) int x d x} d x]+frac{x^{2}}{2}`


`Rightarrow-cos y+[y sin y-int sin y d y]=2[log x times frac{x^{2}}{2}-int frac{1}{x} frac{x^{2}}{2}]+frac{x^{2}}{2}`


`Rightarrow-cos y+y sin y+cos y=x^{2} log x-frac{x^{2}}{2}+frac{x^{2}}{2}+C`


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