The given differential equation is:

\[
y' + \ln(2) \cdot y = 0
\]

This is a first-order linear differential equation, and we can solve it using an integrating factor. Here's the step-by-step solution:

### Step 1: Rewrite the equation in standard form
The standard form of a first-order linear differential equation is:

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

In our case, \( P(x) = \ln(2) \) and \( Q(x) = 0 \). So the equation becomes:

\[
y' + \ln(2) \cdot y = 0
\]

### Step 2: Find the integrating factor
The integrating factor \( \mu(x) \) is given by:

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

Since \( P(x) = \ln(2) \) is constant, the integral becomes:

\[
\mu(x) = e^{\ln(2) \cdot x} = 2^x
\]

### Step 3: Multiply through by the integrating factor
Multiply both sides of the differential equation by the integrating factor \( \mu(x) = 2^x \):

\[
2^x \cdot y' + 2^x \cdot \ln(2) \cdot y = 0
\]

The left-hand side is the derivative of the product \( 2^x \cdot y \), so we can rewrite the equation as:

\[
\frac{d}{dx} \left( 2^x \cdot y \right) = 0
\]

### Step 4: Integrate both sides
Integrating both sides with respect to \( x \):

\[
2^x \cdot y = C
\]

where \( C \) is the constant of integration.

### Step 5: Solve for \( y \)
Finally, solve for \( y \):

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

Thus, the general solution to the differential equation is:

\[
y(x) = \frac{C}{2^x}
\]

### Summary:
The solution to the differential equation \( y' + \ln(2) \cdot y = 0 \) is:

\[
y(x) = \frac{C}{2^x}
\]

where \( C \) is an arbitrary constant.

---

Would you like more details on any of the steps?

Here are some related questions to explore:

1. What if the equation had a non-zero right-hand side (non-homogeneous)?
2. How do we solve first-order linear differential equations with variable coefficients?
3. What is the physical meaning of a differential equation like this in applied contexts?
4. How does the integrating factor method compare to separation of variables?
5. What are the general types of solutions for first-order differential equations?

**Tip**: Integrating factors are powerful for solving linear first-order differential equations, especially when the coefficient of \( y \) is not constant.

This solution explains how to solve the first-order linear differential equation y' + ln(2)·y = 0 using the integrating factor method. The solution includes step-by-step details, important formulas, and a general solution in the form y(x) = C / 2^x. Suitable for high school and university students.

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