To solve the first-order linear differential equation:

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

This is a separable equation. We can solve it step by step.

### Step 1: Rewrite the equation

\[
y' = -\ln(2) \cdot y
\]

This is a separable form where we can move terms involving \(y\) to one side and terms involving \(x\) to the other.

### Step 2: Separate variables

\[
\frac{dy}{y} = -\ln(2) \cdot dx
\]

### Step 3: Integrate both sides

\[
\int \frac{1}{y} \, dy = \int -\ln(2) \, dx
\]

On the left, the integral of \(\frac{1}{y}\) is \(\ln |y|\), and on the right, the integral of \(-\ln(2)\) is \(-\ln(2) \cdot x\).

\[
\ln |y| = -\ln(2) \cdot x + C
\]

### Step 4: Solve for \(y\)

Exponentiate both sides to remove the logarithm:

\[
|y| = e^{-\ln(2) \cdot x + C} = e^{C} \cdot e^{-\ln(2) \cdot x}
\]

Since \(e^C\) is just a constant, we can rewrite it as \(C_1\):

\[
|y| = C_1 e^{-\ln(2) \cdot x}
\]

Thus, \(y\) is:

\[
y = C_1 e^{-\ln(2) \cdot x}
\]

Where \(C_1\) is a constant.

### Final Solution:

\[
y = C_1 e^{-\ln(2) \cdot x}
\]

Would you like more details on any part of the solution?

Here are 5 related questions to explore:
1. How do you solve higher-order linear differential equations?
2. What are the properties of logarithmic functions in differential equations?
3. How does the integrating factor method work for non-separable linear equations?
4. What does the solution of this differential equation represent geometrically?
5. How do initial conditions affect the solution of this type of differential equation?

**Tip:** When solving differential equations, always check if it's separable first—this can often simplify the process.

Learn how to solve the first-order linear differential equation y' + ln(2) * y = 0 using separation of variables and logarithmic integration. This problem covers key concepts in differential equations and logarithmic functions. Suitable for college-level students.

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