We begin by analyzing the given differential equation:

$x(x + y) \frac{dy}{dx} = y(x - y)$

Let's simplify by isolating $\frac{dy}{dx}$:

$\frac{dy}{dx} = \frac{y(x - y)}{x(x + y)}$

At this point, it might be helpful to use a substitution to simplify the equation. Observing the structure of the equation, we see that it suggests a homogeneous form. Let’s set:

$y = vx \quad \Rightarrow \quad \frac{dy}{dx} = v + x \frac{dv}{dx}$

By substituting $y = vx$ into the equation, we get:

$x(x + vx)\left(v + x \frac{dv}{dx}\right) = vx(x - vx)$

Expanding and simplifying each term will allow us to reduce the equation further. Do you want to proceed with this substitution and solve it step-by-step?

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Relative Questions

1. How does the substitution $y = vx$ simplify homogeneous differential equations?
2. What is the significance of converting a differential equation to a separable form?
3. Can all first-order differential equations be solved using substitutions?
4. How can we confirm that a differential equation is homogeneous?
5. What techniques exist for solving non-linear differential equations?

Tip:

In differential equations, substitution can transform a complex equation into a simpler form, often making it easier to solve.