To solve the differential equation \(\frac{dy}{dx} = 4 - \frac{x}{y}\), we can proceed by attempting separation of variables or finding an appropriate substitution. Let's approach the problem systematically.

### Step-by-Step Solution:

1. **Rewrite the equation:**
   \[
   \frac{dy}{dx} = 4 - \frac{x}{y}
   \]

2. **Multiply both sides by \( y \) to eliminate the fraction:**
   \[
   y \frac{dy}{dx} = 4y - x
   \]

3. **This is not easily separable, so we can attempt substitution.** Let us try the substitution:
   \[
   v = \frac{y}{x}
   \quad \Rightarrow \quad y = vx
   \]

4. **Differentiate \( y = vx \) with respect to \( x \):**
   \[
   \frac{dy}{dx} = v + x \frac{dv}{dx}
   \]

5. **Substitute \( y = vx \) and \( \frac{dy}{dx} = v + x \frac{dv}{dx} \) into the original equation:**
   \[
   v + x \frac{dv}{dx} = 4 - \frac{x}{vx}
   \]

   Simplify the right-hand side:
   \[
   v + x \frac{dv}{dx} = 4 - \frac{1}{v}
   \]

6. **Rearrange the equation:**
   \[
   x \frac{dv}{dx} = 4 - \frac{1}{v} - v
   \]

   This is now a separable differential equation.

7. **Separate the variables:**
   \[
   \frac{v}{v^2 - 4v + 1} dv = \frac{dx}{x}
   \]

   At this point, the left-hand side requires partial fraction decomposition to integrate.

Would you like me to proceed with solving the integral step by step?

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### Related Questions:

1. What is the method of substitution in solving differential equations?
2. How does partial fraction decomposition work in solving integrals?
3. Can all differential equations be solved by separation of variables?
4. What are some common substitutions used in solving differential equations?
5. How is integrating factor method different from separation of variables?

#### Tip: Always check if a substitution can simplify complex differential equations.

Learn how to solve the differential equation dy/dx = 4 - x/y through substitution and separation of variables. This detailed solution also involves partial fraction decomposition. Suitable for advanced high school or college-level students studying differential equations.

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