dx/z = dy/-z = dz/z² + (x + y)²

Steps to Solve

1. Identify the Equation First, clearly write down the complex equation involving variables $x$, $y$, and $z$. Let's denote it as an equation representation: $$ \frac{dy}{dx} + P(x)y = Q(x) $$

2. Simplify or Reorganize If applicable, organize the equation into a standard form, where necessary, isolating the derivative and the other terms.

3. Integrate Factors Determine the integrating factor, which is generally $e^{\int P(x)dx}$. Calculate this integrating factor and denote it as $\mu(x)$: $$ \mu(x) = e^{\int P(x)dx} $$

4. Multiply by Integrating Factor Multiply the entire differential equation by the integrating factor, $\mu(x)$, to facilitate integration: $$ \mu(x) \frac{dy}{dx} + \mu(x) P(x) y = \mu(x) Q(x) $$

5. Rewrite as a Derivative Reorganize the left side of the equation into the derivative of a product: $$ \frac{d}{dx} [\mu(x) y] = \mu(x) Q(x) $$

6. Integrate Both Sides Integrate both sides with respect to $x$: $$ \int \frac{d}{dx} [\mu(x) y] dx = \int \mu(x) Q(x) dx $$

7. Solve for y After integrating, isolate $y$. The result will be: $$ y = \frac{1}{\mu(x)} \left( \int \mu(x) Q(x) dx + C \right) $$ where $C$ is the constant of integration.