https://file.aiquickdraw.com/mathbot/file/1726341733089aiuq0dsn.jpg

The given differential equation is:

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

This is a first-order linear differential equation. We can solve it by rewriting it in a more standard form and using an integrating factor method.

### Step-by-step solution:

1. Rewrite the equation in the form \( y' + p(x)y = 0 \).

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

Here, \( p(x) = -\frac{4}{1+x} \).

2. The next step is to find the integrating factor \( \mu(x) \), which is given by:

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

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

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

3. Multiply the entire differential equation by the integrating factor \( (1+x)^{-4} \):

\[
(1+x)^{-4}y' - 4(1+x)^{-5}y = 0
\]

The left-hand side simplifies to the derivative of \( (1+x)^{-4}y \):

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

4. Integrate both sides:

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

where \( C \) is a constant of integration.

5. Solve for \( y \):

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

Thus, the general solution to the differential equation is:

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

Let me know if you would like more details on any part of this process!

Here are some related questions for practice:
1. How do you find the general solution to a first-order linear differential equation?
2. What is an integrating factor, and how is it used in solving differential equations?
3. What conditions would give you a particular solution instead of a general one?
4. How do you handle differential equations with more complicated functions of \(x\)?
5. How does the constant of integration \(C\) affect the family of solutions?

**Tip**: Integrating factors are a powerful technique for solving first-order linear equations, so it's helpful to practice applying them in different scenarios.

Learn how to solve the first-order linear differential equation (1 + x)y' - 4y = 0 using the integrating factor method. This problem covers essential topics in differential equations and is suitable for university-level students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation