To solve the linear differential equation $y' + y = \ln(x)$, follow these steps:

Step 1: Identify the form

This is a first-order linear differential equation in the form: $y' + P(x)y = Q(x)$ where $P(x) = 1$ and $Q(x) = \ln(x)$.

Step 2: Find the integrating factor

The integrating factor $\mu(x)$ is given by: $\mu(x) = e^{\int P(x) \, dx} = e^{\int 1 \, dx} = e^x$

Step 3: Multiply the entire equation by the integrating factor

$e^x y' + e^x y = e^x \ln(x)$

Step 4: Recognize the left-hand side as a derivative

The left-hand side can be written as: $\frac{d}{dx}\left(e^x y\right) = e^x \ln(x)$

Step 5: Integrate both sides

Integrate both sides with respect to $x$: $e^x y = \int e^x \ln(x) \, dx$

Step 6: Solve the integral

The integral $\int e^x \ln(x) \, dx$ can be solved using integration by parts. Let: $u = \ln(x) \quad \text{and} \quad dv = e^x dx$ Then: $du = \frac{1}{x} dx \quad \text{and} \quad v = e^x$ Applying integration by parts: $\int e^x \ln(x) \, dx = e^x \ln(x) - \int e^x \cdot \frac{1}{x} \, dx$ This integral may be left in terms of an integral expression or further simplified if needed, but generally, it is not elementary.

Step 7: Solve for $y(x)$

After integrating, we solve for $y(x)$: $y(x) = e^{-x} \left( e^x \ln(x) - \int e^x \cdot \frac{1}{x} \, dx \right) + C e^{-x}$ where $C$ is the constant of integration.

This is the general solution to the differential equation.

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Would you like a more detailed step-by-step integration process, or do you have any questions about the solution?

Related Questions:

1. How do you apply integration by parts in solving integrals?
2. What are integrating factors and how are they derived?
3. Can all first-order linear differential equations be solved using integrating factors?
4. How does the method change if the differential equation is non-linear?
5. What are the conditions for the existence and uniqueness of solutions to differential equations?

Tip:

When dealing with integrals that are not elementary, it's often helpful to leave them in integral form or use numerical methods if a specific solution is required.