Solving the ODE: dy/dx (xy - y^2) / (x^2 - y^2)

In this guide, we will explore the process of solving the given ordinary differential equation (ODE):
[frac{dy}{dx} frac{xy - y^2}{x^2 - y^2}]

Step 1: Simplifying the Differential Equation

After a bit of algebra, the given ODE can be simplified into a linear first-order equation. Let's see how:

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

By factoring the numerator and the denominator, we can rewrite it as:

[frac{dy}{dx} frac{y(x - y)}{x^2 - y^2}]

Further simplification yields:

[frac{dy}{dx} frac{y}{x - y}]

Step 2: Reciprocating Both Sides

To change the roles of (x) and (y), we reciprocate both sides of the equation. This step is only possible if the ODE does not contain higher-order derivatives like (y^{(n)}) where (n > 1):
[frac{dx}{dy} frac{x - y}{y}]

Step 3: Solving the Linear First-Order Equation

Now that we have a linear first-order differential equation, let's solve it to find the function (xy). The equation is of the form:
[frac{dx}{dy} frac{x - y}{y}]

We can transform this into a linear first-order equation by isolating (x) on one side:

[x - frac{1}{y}x 1]

An integrating factor is required to solve this equation. The integrating factor (mu(y)) is given by:

[mu(y) expleft(int -frac{1}{y} dyright) frac{1}{y}]

Step 4: Applying the Integrating Factor

Multiplying both sides of the linear first-order equation by the integrating factor, we get:

[frac{1}{y}x int frac{1}{y} dy ln|y| - ln|C| ln|frac{C}{y}| ln Cy]

Where (C) is a constant. Rearranging gives:

[frac{x}{y} ln Cy]

Finally, solving for (x), we obtain:

[x y ln Cy]

The General Solution

The general solution of the given ODE is:
[x y ln Cy]

This solution can represent an infinite set of solutions that are valid for this differential equation, where (C) is an arbitary constant.