equation differentielle lineaire y'+y=lnx

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.

Learn how to solve the first-order linear differential equation y' + y = ln(x) using integrating factors and integration by parts. This problem involves advanced concepts in differential equations and requires knowledge of calculus. Suitable for advanced college-level students and mathematicians.

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