solve this eq (x ^ 2 - 2y) * dx - x * dy = 0

To solve the given differential equation:

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

We can approach it as follows:

### Step 1: Rearrange the Equation
Let's rewrite the equation:

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

Now, divide both sides by \( x \cdot (x^2 - 2y) \) to separate variables:

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

### Step 2: Integrate Both Sides
Next, we integrate both sides of the equation.

1. **Integrating the left side**:
   \[
   \int \frac{1}{x} \, dx = \ln |x| + C_1
   \]

2. **Integrating the right side**:
   To integrate the right side, let's consider the substitution \( u = x^2 - 2y \), so \( du = -2 \, dy \). Thus, the integral becomes:

   \[
   \int \frac{dy}{x^2 - 2y} = -\frac{1}{2} \ln |x^2 - 2y| + C_2
   \]

### Step 3: Combine the Integrals
After integrating, we get:

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

Where \( C = C_2 - C_1 \) is a constant.

### Step 4: Simplify the Equation
Exponentiate both sides to remove the logarithms:

\[
|x| = \frac{K}{\sqrt{|x^2 - 2y|}}
\]

Where \( K = e^C \) is a constant.

### Step 5: Final Solution
Rearrange the equation to express the general solution:

\[
x^2(x^2 - 2y) = C
\]

Where \( C \) is an arbitrary constant.

### Final Answer:
The general solution to the differential equation is:

\[
x^2(x^2 - 2y) = C
\]

Do you want further details or have any questions about this solution?

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Here are 5 related questions for further practice:

1. Solve the differential equation \((x^2 + y^2) \, dx - x \, dy = 0\).
2. What is the solution to the differential equation \((x + y) \, dx + (x - y) \, dy = 0\)?
3. Determine the general solution for \(xy \, dx + (x^2 - y^2) \, dy = 0\).
4. How would you solve \((x - y) \, dx + (x + y) \, dy = 0\)?
5. Find the solution to \((x + y)^2 \, dx + (x - y)^2 \, dy = 0\).

**Tip:** When solving separable differential equations, always check if the variables can be separated directly. If not, consider substitutions or integrating factors.

Learn how to solve the separable differential equation (x^2 - 2y) dx - x dy = 0 using the separation of variables method and substitution techniques. This detailed guide walks through each step, from rearranging terms to integrating both sides. Suitable for undergraduate calculus students.

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