https://file.aiquickdraw.com/mathbot/file/1727074607154rfb2wk3r.jpg

The problem in the image asks for the general solution of the differential equation:

\[
y' + y \cdot \cos(x) = 0
\]

### Step-by-step solution:

1. **Rewriting the equation:**

   The differential equation is separable. We can rearrange it as:

   \[
   \frac{dy}{dx} = -y \cdot \cos(x)
   \]

2. **Separation of variables:**

   Separate the variables \( y \) and \( x \):

   \[
   \frac{dy}{y} = -\cos(x) \, dx
   \]

3. **Integration:**

   Integrate both sides:

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

   The integrals are:

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

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

4. **Solving for \( y \):**

   Exponentiate both sides to solve for \( y \):

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

   Let \( e^C = c \), where \( c \) is a constant. Thus, the general solution is:

   \[
   y(x) = c \cdot e^{-\sin(x)}
   \]

### Conclusion:

The correct general solution is:

\[
y(x) = c \cdot e^{-\sin(x)}, \quad c \in \mathbb{R}
\]

This corresponds to the second option in the image:

\[
y(x) = c \cdot e^{-\sin(x)}, \quad c \in \mathbb{R}
\]

---

Would you like more details on this solution or have any questions? Here are some additional questions related to this topic:

1. What is a separable differential equation?
2. How does separation of variables work in solving first-order ODEs?
3. Why is the constant \( c \) introduced when solving differential equations?
4. What other methods can be used to solve first-order linear differential equations?
5. How would the solution change if the equation involved \( \sin(x) \) instead of \( \cos(x) \)?

**Tip:** Always check if a differential equation is separable. This simplifies solving many types of first-order ODEs!

What is a general solution of the differential equation y' + y * cos(x) = 0?

Find the general solution of the first-order differential equation y' + y * cos(x) = 0 using the method of separation of variables and exponential integration. This problem involves core concepts like solving linear ODEs and applying integration techniques.

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