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We are given the equation:

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

The goal is to solve this equation by making an appropriate substitution and expressing the solution in the form \( y^2 = \text{something} \), using \( c \) for the constant of integration.

### Step 1: Substitution
Let's use a substitution to simplify the equation. A common approach is to use polar coordinates. Define a new variable \( z = y/x \), so that \( y = zx \). This transforms the equation into a more manageable form.

Using the substitution \( z = y/x \), we can differentiate \( y = zx \):

\[
dy = z \, dx + x \, dz
\]

Now, substitute \( y = zx \) and \( dy = z \, dx + x \, dz \) into the original equation:

\[
(x^2 + (zx)^2) \, dx + 2x(zx)(z \, dx + x \, dz) = 0
\]

This equation simplifies further, allowing us to solve it step by step. Let me proceed with the simplification and provide the final solution.The solution to the differential equation is:

\[
y(x) = \pm \sqrt{C_1 - 2x^2}
\]

Since the problem asks for the answer in the form \( y^2 = \), we square both sides:

\[
y^2 = C_1 - 2x^2
\]

Here, \( C_1 \) is the constant of integration. We can replace \( C_1 \) with \( c \) to match the problem's notation. Thus, the final solution is:

\[
y^2 = c - 2x^2
\]

Let me know if you'd like further clarification or details! Here are some related questions to extend this problem:

1. How do polar coordinates simplify certain types of differential equations?
2. What are the typical substitution methods for solving non-exact differential equations?
3. How do you determine which constant to introduce in integration?
4. How do boundary conditions influence the form of the solution?
5. What is the geometric interpretation of the solution \( y^2 = c - 2x^2 \)?

**Tip**: Always verify if your substitution makes the equation separable or simplifies the integration process.

Solve (x^2 + y^2) dx + (2xy) dy = 0 by making an appropriate substitution. Enter your answer in the form y^2 = _____, and use c for the unknown constant.

Learn how to solve the first-order differential equation (x^2 + y^2) dx + (2xy) dy = 0 by making an appropriate substitution. The final solution is presented in the form y^2 = c - 2x^2.

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