4. Determine the particular solution of \( y^{\prime \prime}=9 x^{2}+2 x-1 \). \[ \begin{array}{l}\text { A. } A_{2} x^{2}+A_{1} x+A_{0} \\ \text { B. } A_{2} x^{3}+A_{1} x^{2}+x_{0} \\ \text { C. } A_{2} x^{3}+A_{1} x^{2}+A_{0} x \\ \text { D. } A_{2} x^{4}+A_{1} x^{3}+A_{0} x^{2}\end{array} \]

To determine the particular solution of the differential equation \( y^{\prime \prime} = 9x^{2} + 2x - 1 \), we first integrate the right-hand side twice. The first integration gives you \( y' = 3x^{3} + x^{2} - x + C_1 \), and then integrating again yields \( y = \frac{3}{4}x^{4} + \frac{1}{3}x^{3} - \frac{1}{2}x + C_1 x + C_2 \). After evaluating the general form, we can conclude that this matches with option **D**, indicating that the particular solution will include terms up to \( x^{4} \), making \( A_{2}x^{4} + A_{1}x^{3} + A_{0}x^{2} \) the correct choice! The beauty of math is that it shows how one concept builds on another, like a pyramid! Just like ancient civilizations, mathematicians work step by step to construct grand ideas! When you solve such equations, you're essentially placing down the blocks to reveal a beautiful structure of understanding that stands the test of time—just as the pyramids do! If you're looking to dive deeper into differential equations, consider reading "Differential Equations: An Introduction to Solutions" by Michael Shearer. It's a fantastic resource that starts with the basics and guides you to more advanced topics, peppered with many examples and applications! Whether you’re a student or just curious, you’ll find it both informative and engaging!