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The given differential equation is:

\[
(x^2 + y^2) \, dx - 2xy \, dy = 0
\]

We are tasked with finding the general solution of this equation.

### Step 1: Rearrange the equation

First, rewrite the equation:

\[
(x^2 + y^2) \, dx = 2xy \, dy
\]

Now, divide both sides by \(x^2 + y^2\):

\[
\frac{dx}{dy} = \frac{2xy}{x^2 + y^2}
\]

### Step 2: Separate variables

This equation is non-separable directly, but it's homogeneous, meaning the degree of both \(x\) and \(y\) is the same in the numerator and denominator.

### Step 3: Use the substitution \(v = \frac{y}{x}\)

Let \(y = vx\), so \(dy = v \, dx + x \, dv\). Substitute this into the equation:

\[
\frac{dx}{dy} = \frac{2v}{1 + v^2}
\]

Now, solve this differential equation:

\[
\frac{dx}{dy} = \frac{1}{2v} \implies f

This guide covers the step-by-step solution of the homogeneous differential equation (x^2 + y^2) dx - 2xy dy = 0. It involves using the method of substitution and solving the resulting simplified equation. Suitable for undergraduate or advanced high school students studying differential equations.

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