The given differential equation is:

\[
(1 + x)y' - 4y = 0
\]

This is a first-order linear differential equation. To solve it, we follow these steps:

### Step 1: Rewrite the equation in standard form
We need to express the equation in the form:

\[
y' + P(x)y = Q(x)
\]

Dividing the whole equation by \( 1 + x \) (assuming \( 1 + x \neq 0 \)):

\[
y' - \frac{4}{1 + x}y = 0
\]

This is now in standard form with:
- \( P(x) = -\frac{4}{1+x} \)
- \( Q(x) = 0 \)

### Step 2: Find the integrating factor
The integrating factor \( \mu(x) \) is given by:

\[
\mu(x) = e^{\int P(x)\,dx}
\]

Substitute \( P(x) = -\frac{4}{1+x} \):

\[
\mu(x) = e^{\int -\frac{4}{1+x}\,dx} = e^{-4\ln|1+x|}
\]

Simplifying:

\[
\mu(x) = e^{\ln|1+x|^{-4}} = |1+x|^{-4}
\]

### Step 3: Multiply the equation by the integrating factor
Multiply both sides of the equation \( y' - \frac{4}{1+x}y = 0 \) by \( |1+x|^{-4} \):

\[
|1+x|^{-4}y' - \frac{4}{1+x} |1+x|^{-4} y = 0
\]

This simplifies to:

\[
\frac{d}{dx} \left( |1+x|^{-4} y \right) = 0
\]

### Step 4: Integrate both sides
Integrating both sides:

\[
|1+x|^{-4} y = C
\]

Where \( C \) is the constant of integration.

### Step 5: Solve for \( y \)
Now, solve for \( y \):

\[
y = C |1+x|^4
\]

Thus, the general solution to the differential equation is:

\[
y = C (1 + x)^4
\]

### Final Answer:
The general solution is:

\[
y = C (1 + x)^4
\]

Let me know if you want more details or have any questions!

Here are 5 related questions:
1. How would the solution change if the equation was non-homogeneous?
2. What happens if \( 1 + x = 0 \)? How would this affect the solution?
3. Can you solve a similar equation: \( (2 + x)y' - 3y = 0 \)?
4. What is the behavior of the solution as \( x \to \infty \)?
5. How do we solve non-linear first-order differential equations?

**Tip**: Always try to transform differential equations into standard forms to easily apply known methods.

Solve the first-order linear differential equation (1 + x)y′ − 4y = 0 using the integrating factor method. Follow the step-by-step solution covering key concepts in differential equations. Suitable for undergraduate students studying calculus or differential equations.

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