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The equation given is:

\[
y'' + y = \cos x
\]

We are asked if this differential equation admits a periodic solution.

### Step-by-Step Solution:
This is a second-order linear non-homogeneous differential equation. Let's analyze whether it admits a periodic solution.

1. **Homogeneous part**:
   The homogeneous equation is:

   \[
   y'' + y = 0
   \]
   
   The characteristic equation for this is:

   \[
   r^2 + 1 = 0 \implies r = \pm i
   \]

   So, the general solution to the homogeneous equation is:

   \[
   y_h(x) = c_1 \cos(x) + c_2 \sin(x)
   \]
   where \(c_1\) and \(c_2\) are constants.

2. **Particular solution**:
   For the non-homogeneous equation \(y'' + y = \cos x\), we need to find a particular solution.

   Since \(\cos x\) is part of the solution to the homogeneous equation, we will assume the particular solution takes the form:
   
   \[
   y_p(x) = Ax \sin(x) + Bx \cos(x)
   \]
   
   Now, take the first and second derivatives of \(y_p(x)\):

   \[
   y_p'(x) = A(\sin(x) + x \cos(x)) + B(\cos(x) - x \sin(x))
   \]
   
   \[
   y_p''(x) = A(2\cos(x) - x \sin(x)) + B(-2\sin(x) - x \cos(x))
   \]

   Substituting \(y_p(x)\), \(y_p'(x)\), and \(y_p''(x)\) into the original equation \(y'' + y = \cos(x)\), after simplifying, we can determine the coefficients \(A\) and \(B\).

3. **Conclusion**:
   The homogeneous solution contains the periodic functions \(\cos(x)\) and \(\sin(x)\), but the particular solution includes terms involving \(x\), such as \(x \sin(x)\) and \(x \cos(x)\), which are not periodic. Since the particular solution grows without bound as \(x\) increases (because of the \(x\)-terms), the general solution, which is the sum of the homogeneous and particular solutions, is not periodic.

Thus, **the equation does not admit a periodic solution**.

Would you like more details on any part of the process or have any questions?

### Related questions:
1. What is the general solution for non-homogeneous linear differential equations?
2. Can a non-periodic forcing function lead to a periodic solution?
3. How does resonance in differential equations affect periodic solutions?
4. What are the steps to find a particular solution using the method of undetermined coefficients?
5. How do we classify different types of second-order differential equations?

**Tip:** When analyzing the periodicity of a differential equation solution, always examine both the homogeneous and particular parts carefully.

Does the equation y'' + y = cos(x) admit a periodic solution?

Explore whether the differential equation y'' + y = cos(x) admits a periodic solution. This problem covers key concepts in differential equations, including the method of undetermined coefficients and the analysis of periodicity. Suitable for students at the undergraduate level.

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