The general solution of the differential equation $x \cos y \mathrm{~d} y=\left(x \mathrm{e}^x \log…
The general solution of the differential equation $x \cos y \mathrm{~d} y=\left(x \mathrm{e}^x \log x+\mathrm{e}^x\right) \mathrm{d} x$ is given by
$\sin y=\mathrm{e}^x+\operatorname{cog} x$, where c is a constant of integration.
$\quad \sin y=\mathrm{e}^x \log x+\mathrm{c}$, where c is a constant of integration.
$\mathrm{e}^x \sin y=\log x+\mathrm{c}$, where c is a constant of integration,
$\sin y=\mathrm{ce}^x+\log x$, where c is a constant of integration.
Solution
$\begin{aligned}
& x \cos y \mathrm{~d} y=\left(x \mathrm{e}^x \log x+\mathrm{e}^x\right) \mathrm{d} x \\
& \Rightarrow \cos y \mathrm{~d} y=\mathrm{e}^x\left(\log x+\frac{1}{x}\right) \mathrm{d} x
\end{aligned}$
Integrating on both sides, we get
$\sin y=\mathrm{e}^x \log x+\mathrm{c}$