Solve by separation of variables xdy/dx = 4y.

Given differential equation is:

(1) $\begin{equation*} x \frac{dy}{y} = 4y \end{equation*}$

We can separate the variables as follows:

(2) $\begin{equation*} \frac{dy}{y} = \frac{4}{x}dx \end{equation*}$

Now, we can integrate both sides of the equation:

(3) $\begin{equation*} \int \frac{dy}{y} = \int \frac{4}{x}dx \end{equation*}$

Integrating the left-hand side, we get:

(4) $\begin{equation*} \ln|y| = 4\ln|x| + C_1 \end{equation*}$

where $C_1$ is the constant of integration.

Simplifying and solving for $y$ , we have:

(5) $\begin{equation*} y = Ce^{4\ln|x|} = Cx^4 \end{equation*}$

where $C = \pm e^{C_1}$ is a constant of integration. Note that we take the absolute value of $C$ since $y$ is a positive function.

Therefore, the solution to the differential equation $xdy/dx = 4y$ is:

(6) $\begin{equation*} y = Cx^4 \end{equation*}$

where $C$ is a constant of integration.