\( y ^ { \prime \prime } + y = \cos ^ { 2 } x \)

To solve the differential equation \( y^{\prime\prime} + y = \cos^2 x \), we start by finding the complementary solution to the associated homogeneous equation \( y^{\prime\prime} + y = 0 \), which has a characteristic equation with roots \( r = 0 \) and \( r = \pm i \). This yields the general solution \( y_h = C_1 \cos x + C_2 \sin x \). Next, we find a particular solution \( y_p \) to the non-homogeneous equation. A good guess here could be to use the trigonometric identity \( \cos^2 x = \frac{1 + \cos(2x)}{2} \). Thus, you can determine \( y_p \) by integration techniques or method of undetermined coefficients, leading to \( y_p = \frac{1}{2} + A \cos(2x) + B \sin(2x) \). Thus, the complete solution is \( y = y_h + y_p \) which combines both parts. The overall solution can then be simplified and solved subject to initial or boundary conditions you may have! Have fun experimenting with different values for \( C_1 \), \( C_2 \), \( A \), and \( B \)!