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The given problem is:

\[
(2x - 5y) \, dx + (4x - y) \, dy = 0, \quad y(1) = 4.
\]

This is a first-order differential equation. To solve it, we can check if it’s exact by verifying if the partial derivatives satisfy the condition for exactness. The condition for exactness is that for an equation \( M(x, y) \, dx + N(x, y) \, dy = 0 \), it is exact if \( \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x} \).

### Step 1: Define \( M(x, y) \) and \( N(x, y) \)
In this case:
- \( M(x, y) = 2x - 5y \)
- \( N(x, y) = 4x - y \)

### Step 2: Check for Exactness
Calculate \( \frac{\partial M}{\partial y} \) and \( \frac{\partial N}{\partial x} \):
- \( \frac{\partial M}{\partial y} = -5 \)
- \( \frac{\partial N}{\partial x} = 4 \)

Since \( \frac{\partial M}{\partial y} \neq \frac{\partial N}{\partial x} \), the equation is **not exact**.

### Step 3: Find an Integrating Factor
Since the equation is not exact, we need an integrating factor to make it exact. Commonly, integrating factors are functions of \( x \) or \( y \) alone. 

#### Trying \( \mu(x) = x \) as an integrating factor
Multiply the entire equation by \( x \):
\[
x(2x - 5y) \, dx + x(4x - y) \, dy = 0.
\]
This simplifies to:
\[
(2x^2 - 5xy) \, dx + (4x^2 - xy) \, dy = 0.
\]

Now let:
- \( M(x, y) = 2x^2 - 5xy \)
- \( N(x, y) = 4x^2 - xy \)

Check for exactness again:
- \( \frac{\partial M}{\partial y} = -5x \)
- \( \frac{\partial N}{\partial x} = 4x - y =

(2x - 5y) dx + (4x - y) dy = 0, y(1) = 4.

Find the solution to the first-order differential equation (2x - 5y) dx + (4x - y) dy = 0 with initial condition y(1) = 4. This problem explores the concepts of exact equations and integrating factors, providing a detailed step-by-step solution.

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