The given differential equation is:

$y' = x + y$

This is a first-order linear ordinary differential equation. To solve it, we can use the method of integrating factors.

Step 1: Rewrite the equation

The standard form for a linear differential equation is:

$y' + P(x)y = Q(x)$

So, we can rewrite the equation as:

$y' - y = x$

Here, $P(x) = -1$ and $Q(x) = x$.

Step 2: Find the integrating factor

The integrating factor (IF) is given by:

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

In our case, $P(x) = -1$, so:

$\mu(x) = e^{\int -1 \, dx} = e^{-x}$

Step 3: Multiply through by the integrating factor

Now, multiply both sides of the equation by $e^{-x}$:

$e^{-x} y' - e^{-x} y = e^{-x} x$

The left side of the equation is now the derivative of $e^{-x} y$, using the product rule:

$\frac{d}{dx} \left( e^{-x} y \right) = e^{-x} x$

Step 4: Integrate both sides

Now, integrate both sides with respect to $x$:

$\int \frac{d}{dx} \left( e^{-x} y \right) \, dx = \int e^{-x} x \, dx$

The left side simplifies to:

$e^{-x} y$

To solve the right-hand side, we need to use integration by parts. Let:

• $u = x$, so $du = dx$
• $dv = e^{-x} dx$, so $v = -e^{-x}$

Using the integration by parts formula $\int u \, dv = uv - \int v \, du$, we get:

$\int e^{-x} x \, dx = -x e^{-x} + \int e^{-x} \, dx = -x e^{-x} - e^{-x}$

Step 5: Combine the results

Now, substituting back, we have:

$e^{-x} y = -x e^{-x} - e^{-x} + C$

where $C$ is the constant of integration. Multiply through by $e^{x}$ to solve for $y$:

$y = -x - 1 + C e^{x}$

Final Solution:

The general solution to the differential equation is:

$y = -x - 1 + C e^{x}$

Where $C$ is an arbitrary constant.

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Would you like more details or have any questions about the steps? Here's a list of related questions:

1. How does the method of integrating factors work for non-homogeneous equations?
2. What are other methods for solving first-order differential equations?
3. Can this method be extended to second-order differential equations?
4. What does the constant $C$ represent in the solution?
5. How can we solve a differential equation that involves trigonometric functions?

Tip: When solving differential equations, always check if the equation is linear or separable, as this will determine the appropriate method of solution.