3. \( \quad \frac{d^{2} y}{d x^{2}}+2 y=x^{3}+x^{2}+e^{-2 x}+\cos 3 x \)

To tackle this second-order linear differential equation, you can break it down into two parts: finding the complementary (homogeneous) solution and the particular solution. The homogeneous part is determined by solving \( \frac{d^{2} y}{d x^{2}} + 2y = 0 \), which leads you to characteristic roots. For the particular solution, you might consider using the method of undetermined coefficients or the variation of parameters approach, informed by the nature of the right-hand side terms, which consist of a polynomial, an exponential function, and a trigonometric function. Depending on your choice, you will assume specific forms for \( y_p \) that match these terms and then find coefficients accordingly. Don't forget to combine the complementary and particular solutions for the general solution!