To solve the differential equation 

\[
\frac{dy}{dx} = \frac{x + 3y}{3x + y},
\]

we can use the substitution method. Let's set up \( v = \frac{y}{x} \), where \( y = vx \), and find a solution by reducing the equation to a separable form.

### Step 1: Substitute \( y = vx \)
Let \( y = vx \), where \( v \) is a function of \( x \), which implies:
\[
\frac{dy}{dx} = v + x \frac{dv}{dx}.
\]

### Step 2: Substitute into the Differential Equation
Replacing \( y \) and \( \frac{dy}{dx} \) in the original equation gives:
\[
v + x \frac{dv}{dx} = \frac{x + 3(vx)}{3x + v x}.
\]

Simplify the right side:
\[
v + x \frac{dv}{dx} = \frac{x(1 + 3v)}{x(3 + v)}.
\]

Cancel \( x \) from both the numerator and denominator on the right:
\[
v + x \frac{dv}{dx} = \frac{1 + 3v}{3 + v}.
\]

### Step 3: Rearrange to Solve for \( \frac{dv}{dx} \)
Now, isolate \( x \frac{dv}{dx} \):
\[
x \frac{dv}{dx} = \frac{1 + 3v}{3 + v} - v.
\]

Combine terms on the right:
\[
x \frac{dv}{dx} = \frac{1 + 3v - v(3 + v)}{3 + v}.
\]

Expanding and simplifying inside the numerator:
\[
x \frac{dv}{dx} = \frac{1 + 3v - 3v - v^2}{3 + v},
\]
\[
x \frac{dv}{dx} = \frac{1 - v^2}{3 + v}.
\]

### Step 4: Separate Variables
Now, separate the variables \( v \) and \( x \):
\[
\frac{3 + v}{1 - v^2} dv = \frac{dx}{x}.
\]

This equation is now separable and can be integrated to find \( v(x) \), and ultimately \( y(x) \).

Would you like to see the integration steps? 

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1. What is the separable method for solving differential equations?
2. How does the substitution \( y = vx \) simplify solving differential equations?
3. How do we isolate \( \frac{dv}{dx} \) in this problem?
4. What other methods exist for solving non-linear differential equations?
5. How do substitution methods help in reducing complex differential equations?

**Tip:** When you encounter \(\frac{dy}{dx}\) in an equation, think about substitutions that can simplify the variables into a more manageable form.

Explore the solution to the differential equation dy/dx = (x + 3y) / (3x + y) by using substitution (y = vx) and separation of variables. This step-by-step solution covers methods for simplifying complex differential equations, making it suitable for college-level studies.

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