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The given differential equation is:

\[
y'' + y' + y = \sin^2(x)
\]

This is a non-homogeneous second-order linear differential equation. To solve it, we typically follow these steps:

### 1. Solve the homogeneous equation
The homogeneous version of the equation is:

\[
y'' + y' + y = 0
\]

The characteristic equation associated with this is:

\[
r^2 + r + 1 = 0
\]

Solving this quadratic equation:

\[
r = \frac{-1 \pm \sqrt{1^2 - 4(1)(1)}}{2(1)} = \frac{-1 \pm \sqrt{-3}}{2}
\]

\[
r = \frac{-1 \pm \sqrt{3}i}{2}
\]

Thus, the roots are complex numbers:

\[
r = \frac{-1}{2} \pm \frac{\sqrt{3}}{2}i
\]

Therefore, the solution to the homogeneous equation is:

\[
y_h(x) = e^{-\frac{x}{2}} \left( C_1 \cos\left(\frac{\sqrt{3}}{2}x\right) + C_2 \sin\left(\frac{\sqrt{3}}{2}x\right) \right)
\]

### 2. Find the particular solution
To find the particular solution, we use the method of undetermined coefficients. The right-hand side of the equation is \(\sin^2(x)\). We can rewrite this using a trigonometric identity:

\[
\sin^2(x) = \frac{1 - \cos(2x)}{2}
\]

Thus, the equation becomes:

\[
y'' + y' + y = \frac{1}{2} - \frac{1}{2}\cos(2x)
\]

We will assume the particular solution is of the form:

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

Substitute \(y_p(x)\) and its derivatives into the original equation, then solve for \(A\), \(B\), and \(C\).

### 3. General solution
The general solution to the differential equation is:

\[
y(x) = y_h(x) + y_p(x)
\]

Would you like a detailed solution for the particular part or any other clarifications?

### Related questions:
1. What is the general form of solving non-homogeneous differential equations?
2. How are trigonometric identities useful in solving differential equations?
3. Why do we separate the homogeneous and particular solutions in differential equations?
4. How do complex roots of a characteristic equation affect the solution form?
5. How do the method of undetermined coefficients and variation of parameters differ?

**Tip**: Always simplify the right-hand side using trigonometric identities when dealing with sine or cosine terms in differential equations.

Explore the solution to the non-homogeneous differential equation y'' + y' + y = sin^2(x). This step-by-step guide explains how to solve the equation using trigonometric identities and the method of undetermined coefficients, including finding both the homogeneous and particular solutions. Ideal for university-level students studying differential equations.

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