Example 4.4. Find the general solution of the first-order differential equation $x y^{\prime}-\frac{y}{\ln x}=0$.
Solution. Express the derivative $y^{\prime}$ from the equation $y^{\prime}=\frac{y}{x \ln x}$. The right-hand side can be factored as $f(x, y)=\phi(x) \cdot \psi(y)$, hence this is a separable differential equation. Separate the variables $\frac{d y}{d x}=\frac{y}{x \ln x}$ or $\frac{d y}{y}=\frac{d x}{x \ln x}$. Now integrate both sides and for convenience, write the constant of integration as $\ln c$.

\[
\begin{aligned}
& \int \frac{d y}{y}=\int \frac{d x}{x \ln x}, \ln |y|=\int \frac{d \ln x}{\ln x}, \\
& \ln |y|=\ln |\ln x|+\ln c(c>0)
\end{aligned}
\]

or

\[
\ln |y|=\ln c|\ln x|,
\]

then

\[
|y|=\mathrm{c}|\ln x| .
\]

Finally, $y=c \cdot \ln x$, where $c$ is any number. Thus, the general solution is $y=c \ln x$.