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.

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