To make the differential equation \((2x - 1) \, dx + (3y + 7) \, dy = 0\) exact, we need to adjust it so that it satisfies the exactness condition. An exact differential equation is of the form:

\[
M(x, y) \, dx + N(x, y) \, dy = 0,
\]

where \(M(x, y)\) and \(N(x, y)\) satisfy the condition:

\[
\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}.
\]

### Step 1: Define \(M(x, y)\) and \(N(x, y)\)
From the given equation:

\[
M(x, y) = 2x - 1, \quad N(x, y) = 3y + 7.
\]

### Step 2: Check for Exactness
Compute the partial derivatives:

\[
\frac{\partial M}{\partial y} = \frac{\partial}{\partial y} (2x - 1) = 0,
\]
\[
\frac{\partial N}{\partial x} = \frac{\partial}{\partial x} (3y + 7) = 0.
\]

Since both partial derivatives are zero, the equation is already exact. However, we need to apply an integrating factor to ensure this relationship holds if the derivatives were not initially equal.

### Step 3: Find an Integrating Factor
If the equation were not exact, we would find an integrating factor that depends either on \(x\) or \(y\). An integrating factor \(\mu(x, y)\) can make the equation exact by multiplying it throughout:

\[
\mu(x, y) \cdot (2x - 1) \, dx + \mu(x, y) \cdot (3y + 7) \, dy = 0.
\]

For this equation, common approaches involve integrating factors of the form \(\mu(x)\) or \(\mu(y)\).

#### Example Integrating Factor (if needed)
If the equation weren't exact and an integrating factor of the form \(\mu(x) = x\) was appropriate, it would multiply through to give:

\[
x(2x - 1) \, dx + x(3y + 7) \, dy = 0.
\]

However, since the equation \((2x - 1) \, dx + (3y + 7) \, dy = 0\) already satisfies the condition of exactness, no further steps are necessary.

Would you like further assistance or additional information?

### Related Questions:
1. What are the common types of integrating factors used for non-exact equations?
2. How to solve exact differential equations after finding the integrating factor?
3. What are the steps to solve non-exact differential equations systematically?
4. How can we determine whether an equation is already exact or needs modification?
5. What are examples of differential equations that require integrating factors involving both \(x\) and \(y\)?

**Tip:** Checking exactness first can save time—if it’s exact, integration can be applied directly to find the solution!

Learn how to make the differential equation (2x - 1) dx + (3y + 7) dy exact with a detailed step-by-step solution. Understand the exactness condition, integrating factors, and practical examples. Suitable for advanced undergraduate students and enthusiasts of differential equations.

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