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The differential equation provided is:

\[
(x^2 + 3xy + y^2)dx - x^2 dy = 0
\]

### Step 1: Rewrite the equation
The given differential equation can be rewritten as:

\[
(x^2 + 3xy + y^2) \, dx = x^2 \, dy
\]

Or:

\[
\frac{dy}{dx} = \frac{x^2 + 3xy + y^2}{x^2}
\]

### Step 2: Simplify the equation
Let's simplify the right-hand side:

\[
\frac{dy}{dx} = 1 + \frac{3y}{x} + \frac{y^2}{x^2}
\]

This can be rewritten as:

\[
\frac{dy}{dx} = \left(\frac{y}{x} + 1\right)^2
\]

### Step 3: Use substitution
Let \( v = \frac{y}{x} \), which implies \( y = vx \). Therefore, the derivative \( \frac{dy}{dx} \) becomes:

\[
\frac{dy}{dx} = v + x\frac{dv}{dx}
\]

Substitute this into the differential equation:

\[
v + x\frac{dv}{dx} = (v + 1)^2
\]

### Step 4: Separate variables
Rearrange the equation to separate variables \( v \) and \( x \):

\[
x \frac{dv}{dx} = (v + 1)^2 - v
\]

Simplifying further:

\[
x \frac{dv}{dx} = v^2 + 2v + 1 - v = v^2 + v + 1
\]

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

### Step 5: Integrate both sides
Integrate both sides:

\[
\int \frac{dv}{v^2 + v + 1} = \int \frac{dx}{x}
\]

The integral on the right is straightforward:

\[
\ln|x| + C_1
\]

For the integral on the left, let's solve it. We complete the square for the quadratic expression \( v^2 + v + 1 \):

\[
v^2 + v + 1 = \left(v + \frac{1}{2}\right)^2 + \frac{3}{4}
\]

This allows us to integrate using the substitution \( u = v + \frac{1}{2} \):

\[
\int \frac{dv}{v^2 + v + 1} = \frac{2}{\sqrt{3}} \arctan \left(\frac{2v + 1}{\sqrt{3}}\right) + C_2
\]

### Step 6: Write the general solution
Finally, substituting back \( v = \frac{y}{x} \) and combining the constants \( C_1 \) and \( C_2 \) into a single constant \( C \), the general solution is:

\[
\frac{2}{\sqrt{3}} \arctan \left(\frac{2y/x + 1}{\sqrt{3}}\right) = \ln|x| + C
\]

This equation describes the general solution to the differential equation provided.

Would you like further details or have any questions? Here are some related topics you might find interesting:

1. How to solve homogeneous differential equations.
2. Methods for integrating functions involving trigonometric substitutions.
3. The process of completing the square in solving integrals.
4. Techniques for solving first-order linear differential equations.
5. The concept and application of substitution in differential equations.
6. Solutions of differential equations using separable variables.
7. General solutions versus particular solutions in differential equations.
8. The role of arbitrary constants in the general solution of differential equations.

**Tip:** When solving differential equations, always look for opportunities to simplify or factor expressions early in the process; this can make integration and further steps much easier.

Learn how to solve the differential equation (x^2 + 3xy + y^2)dx - x^2 dy = 0 with detailed step-by-step instructions. This problem covers differential equations involving separable variables and integrals. Suitable for advanced undergraduate students.

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