We are given the differential equation:

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

We are tasked with finding \( y^2 \). Let's solve this step by step.

### Step 1: Separate variables

We will first try to separate the variables \( x \) and \( y \). To do this, divide both sides of the equation by \( y \):

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

Multiply both sides by \( y \) to group all the \( y \)-terms on one side:

\[
y(9 + x^2) \frac{dy}{dx} = x
\]

Now, rearrange the equation to separate the variables:

\[
y \, dy = \frac{x}{9 + x^2} \, dx
\]

### Step 2: Integrate both sides

We will now integrate both sides. Starting with the left-hand side:

\[
\int y \, dy = \frac{y^2}{2}
\]

Now, let's handle the right-hand side:

\[
\int \frac{x}{9 + x^2} \, dx
\]

This can be solved using a substitution. Let:

\[
u = 9 + x^2 \implies du = 2x \, dx
\]

Thus:

\[
\int \frac{x}{9 + x^2} \, dx = \frac{1}{2} \int \frac{du}{u} = \frac{1}{2} \ln |u| = \frac{1}{2} \ln |9 + x^2|
\]

### Step 3: Write the general solution

Now, putting the results of the integration together:

\[
\frac{y^2}{2} = \frac{1}{2} \ln |9 + x^2| + C
\]

Multiply through by 2 to eliminate the fraction:

\[
y^2 = \ln |9 + x^2| + 2C
\]

Since \( 2C \) is just a constant, we can rename it as a new constant \( C' \):

\[
y^2 = \ln |9 + x^2| + C'
\]

This is the general solution for \( y^2 \).

Would you like more details or have any questions? 

Here are 5 related questions:
1. How does separation of variables work for solving differential equations?
2. Can you explain the substitution method for integrating?
3. What is the general solution of a first-order differential equation?
4. How would we handle an initial condition if given?
5. Why is it valid to rename constants during integration?

**Tip:** Always check if the integration needs a substitution to simplify complex terms before solving!

Learn how to solve the differential equation (9 + x^2) dy/dx = x/y using separation of variables and integration by substitution. This solution covers important concepts in differential equations and calculus, suitable for undergraduate students in Calculus I/II.

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