The given equation is:

$(2x + y - 1) \, dy = (x - 2y + 5) \, dx$

This is a first-order differential equation. To solve it, we aim to separate the variables or recognize any special type of equation (e.g., exact, homogeneous, or linear).

First, let's rewrite the equation as:

$\frac{dy}{dx} = \frac{x - 2y + 5}{2x + y - 1}$

Now we can check if it's exact or separable.

Step 1: Check if the equation is exact

For an exact equation, we would have the form $M(x, y) \, dx + N(x, y) \, dy = 0$, where:

$M(x, y) = x - 2y + 5 \quad \text{and} \quad N(x, y) = -(2x + y - 1)$

To check if it's exact, we find the partial derivatives:

$\frac{\partial M}{\partial y} = -2 \quad \text{and} \quad \frac{\partial N}{\partial x} = -2$

Since $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$, the equation is exact.

Step 2: Solve the exact equation

Since the equation is exact, there exists a function $\psi(x, y)$ such that:

$\frac{\partial \psi}{\partial x} = M(x, y) = x - 2y + 5 \quad \text{and} \quad \frac{\partial \psi}{\partial y} = N(x, y) = -(2x + y - 1)$

Integrating $M(x, y)$ with respect to $x$:

$\psi(x, y) = \int (x - 2y + 5) \, dx = \frac{x^2}{2} - 2xy + 5x + h(y)$ where $h(y)$ is a function of $y$ that appears because of the partial integration.

Differentiate $\psi(x, y)$ with respect to $y$:

$\frac{\partial \psi}{\partial y} = -2x + h'(y)$

Now, set this equal to $N(x, y)$: $-2x + h'(y) = -(2x + y - 1)$

Simplifying: $-2x + h'(y) = -2x - y + 1$

Thus: $h'(y) = -y + 1$

Integrating $h'(y)$ with respect to $y$:

$h(y) = -\frac{y^2}{2} + y + C$

Step 3: Write the solution

The general solution is given by:

$\psi(x, y) = \frac{x^2}{2} - 2xy + 5x - \frac{y^2}{2} + y + C = 0$

This is the implicit solution to the differential equation.

Would you like more details or have any questions?

Here are five related questions to explore:

1. How do we determine if a differential equation is separable?
2. What other methods can be used to solve non-exact differential equations?
3. What are the conditions for a differential equation to be homogeneous?
4. Can you derive a differential equation from real-life applications like population models?
5. How can we verify that a solution satisfies a differential equation?

Tip: When solving exact equations, always check that the mixed partial derivatives of the functions are equal.