The given differential equation is:

\[
y' + y = e^x
\]

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

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

Here, \( P(x) = 1 \) and \( Q(x) = e^x \).

To solve this, we use the method of integrating factors. The integrating factor \( \mu(x) \) is given by:

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

Now, multiply both sides of the original equation by \( e^x \):

\[
e^x y' + e^x y = e^x e^x
\]

This simplifies to:

\[
\frac{d}{dx} \left( e^x y \right) = e^{2x}
\]

Next, integrate both sides with respect to \( x \):

\[
e^x y = \int e^{2x} \, dx
\]

The right-hand side is:

\[
\int e^{2x} \, dx = \frac{1}{2} e^{2x} + C
\]

Thus, we have:

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

Finally, solve for \( y \) by dividing both sides by \( e^x \):

\[
y = \frac{1}{2} e^x + Ce^{-x}
\]

This is the general solution to the differential equation.

Would you like a more detailed explanation of any steps?

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Here are some related questions:
1. How do you apply the integrating factor method to non-homogeneous equations?
2. How does the solution change if \( Q(x) \) is a different function?
3. What is the significance of the constant \( C \) in the solution?
4. How can we verify the solution by substituting it back into the original equation?
5. How can the method of integrating factors be extended to higher-order differential equations?

**Tip:** Always check your solution by substituting it back into the original differential equation to ensure it satisfies the equation.

Explore how to solve the first-order linear differential equation y' + y = e^x using the integrating factor method. This detailed guide covers key steps, including finding the integrating factor, applying integration, and solving for the general solution. Ideal for college-level students studying calculus or differential equations.

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