(x + 1)
dy
dx
 + (x + 2)y = 8xe−x. Give the largest interval I over which the general solution is defined. Determine whether there are any transient terms in the general solution

Given the differential equation:

\[
(x + 1) \frac{dy}{dx} + (x + 2)y = 8xe^{-x}
\]

We need to determine the largest interval \( I \) over which the general solution is defined and identify any transient terms in the solution.

### Step 1: Simplify and solve the differential equation

This is a first-order linear differential equation of the form:

\[
\frac{dy}{dx} + P(x)y = Q(x)
\]

We first rewrite the equation in the standard form by dividing the entire equation by \( x + 1 \):

\[
\frac{dy}{dx} + \frac{x + 2}{x + 1} y = \frac{8xe^{-x}}{x + 1}
\]

Here, \( P(x) = \frac{x + 2}{x + 1} \) and \( Q(x) = \frac{8xe^{-x}}{x + 1} \).

### Step 2: Find the integrating factor

The integrating factor \( \mu(x) \) is given by:

\[
\mu(x) = e^{\int P(x) \, dx} = e^{\int \frac{x + 2}{x + 1} \, dx}
\]

Simplifying \( \frac{x + 2}{x + 1} \):

\[
\frac{x + 2}{x + 1} = 1 + \frac{1}{x + 1}
\]

Thus:

\[
\mu(x) = e^{\int \left(1 + \frac{1}{x + 1}\right) dx} = e^{x + \ln|x + 1|} = e^x \cdot |x + 1|
\]

So the integrating factor is:

\[
\mu(x) = |x + 1| e^x
\]

### Step 3: Multiply the equation by the integrating factor

Multiplying the original equation by the integrating factor:

\[
|x + 1| e^x \frac{dy}{dx} + |x + 1| e^x \frac{x + 2}{x + 1} y = |x + 1| e^x \frac{8xe^{-x}}{x + 1}
\]

Simplifying:

\[
\frac{d}{dx}\left(|x + 1| e^x y\right) = 8x
\]

### Step 4: Integrate both sides

Integrating both sides with respect to \( x \):

\[
|x + 1| e^x y = \int 8x \, dx
\]

\[
|x + 1| e^x y = 4x^2 + C
\]

Therefore:

\[
y = \frac{4x^2 + C}{|x + 1| e^x}
\]

### Step 5: Determine the interval \( I \)

The solution \( y(x) \) is defined as long as \( x + 1 \neq 0 \). Hence, the function is undefined at \( x = -1 \). The largest interval over which the solution is defined depends on where \( x + 1 \) is nonzero.

Thus, the largest interval \( I \) is:

\[
I = (-\infty, -1) \cup (-1, \infty)
\]

### Step 6: Identify transient terms

Transient terms are components of the solution that tend to zero as \( x \) increases (i.e., as \( x \rightarrow \infty \)).

Here, the term involving the constant \( C \):

\[
\frac{C}{|x + 1| e^x}
\]

is a transient term because \( |x + 1| e^x \) grows exponentially as \( x \rightarrow \infty \), causing the term \( \frac{C}{|x + 1| e^x} \) to approach zero.

### Conclusion

- **Largest interval \( I \):** The general solution is defined on \( I = (-\infty, -1) \cup (-1, \infty) \).
- **Transient terms:** The term \( \frac{C}{|x + 1| e^x} \) is a transient term in the solution.

Would you like further details or have any questions?

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**Related Questions:**

1. How do you identify the integrating factor for non-linear equations?
2. What is the significance of transient terms in differential equations?
3. How can we determine the stability of the solutions of a differential equation?
4. What is the difference between homogeneous and non-homogeneous differential equations?
5. Can there be more than one integrating factor for a given differential equation?

**Tip:** In linear differential equations, always check for singular points, as they can affect the domain of the solution.

Explore how to solve a first-order linear differential equation involving an integrating factor and identify transient terms. This problem involves finding the largest interval where the solution is defined and analyzing its behavior. Suitable for advanced undergraduate students and learners of differential equations.

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