Question

Solve $x\cos ydy=(x{e}^{x}Inx+e^x)dx$

Answer 1

$x\cos ydy=(x{e}^x\ln x+e^x)dx\Rightarrow x\cos ydy=x(e^x\ln x+\frac{e^x}{x})dx\Rightarrow \cos ydy=(e^x\ln x+\frac{e^x}{x})dx$

Integrating both sides we get

$\Rightarrow \int \cos ydy=\int (e^x\ln x+\frac{e^x}{x})dx\Rightarrow \int \cos ydy=\int \left(e^x\ln x\right)dx+\int \left(\frac{e^x}{x}\right)dx\Rightarrow \int \cos ydy=\ln x\int e^{x}dx-\int e^x.\frac{1}{x}dx+\int \frac{e^x}{x}dx\Rightarrow \int \cos ydy=\ln x\int e^{x}dx\Rightarrow \sin y=e^x\ln x+e\Rightarrow y={\sin }^{-1}(e^x\ln x+e)$