The equation provided in the image is:

$(x^2 - y^2) \, dx = 2xy \, dy$

This is a first-order differential equation. To solve it, we'll try to separate the variables or use an appropriate method to integrate it.

Step 1: Rewrite the equation

We can rewrite the given equation by moving all terms involving $y$ to one side and all terms involving $x$ to the other:

$\frac{x^2 - y^2}{2xy} \, dx = \, dy$

Step 2: Simplify the equation

Simplify the expression on the left-hand side:

$\frac{x^2}{2xy} - \frac{y^2}{2xy} \, dx = \, dy$

$\frac{x}{2y} - \frac{y}{2x} \, dx = \, dy$

Step 3: Separate the variables

Now, separate the variables $x$ and $y$:

$\left( \frac{x}{2y} - \frac{y}{2x} \right) dx = dy$

$\frac{x}{2y} \, dx - \frac{y}{2x} \, dx = dy$

We can multiply the entire equation by $2$ to simplify it:

$\frac{x}{y} \, dx - \frac{y}{x} \, dx = 2 \, dy$

Step 4: Integrate both sides

To solve this, integrate both sides of the equation.

Integrate the first term:

$\int \frac{x}{y} \, dx = \frac{1}{y} \int x \, dx = \frac{1}{y} \cdot \frac{x^2}{2}$

Integrate the second term:

$\int -\frac{y}{x} \, dx = -y \int \frac{1}{x} \, dx = -y \ln |x|$

Integrate the right-hand side:

$\int 2 \, dy = 2y$

So we have:

$\frac{x^2}{2y} - y \ln |x| = 2y + C$

Where $C$ is the constant of integration.

Final Answer

Thus, the general solution to the differential equation is:

$\frac{x^2}{2y} - y \ln |x| = 2y + C$

Would you like any further explanation or have any questions?

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Related Questions:

1. What are the common techniques for solving first-order differential equations?
2. How do you determine when to use separation of variables for solving a differential equation?
3. What is the method of integrating factors, and how does it apply to linear differential equations?
4. Can all differential equations be solved analytically, or do some require numerical methods?
5. How do initial conditions affect the solution of a differential equation?

Tip: When solving differential equations, always consider simplifying the equation first to make it easier to separate variables or apply other techniques.