4. Determine the particular solution of \( y^{\prime \prime}=9 x^{2}+2 x-1 \) A. \( \boldsymbol{A}_{2} x^{2}+\boldsymbol{A}_{1} x+\boldsymbol{A}_{0} \) B. \( A_{2} x^{3}+A_{1} x^{2}+x_{0} \) C. \( A_{2} x^{3}+A_{1} x^{2}+A_{0} x \) D. \( \mathbf{A}_{2} \mathbf{x}^{4}+\mathbf{A}_{1} \mathbf{x}^{3}+\mathbf{A}_{0} \mathbf{x}^{2} \)

To find the particular solution of the differential equation \( y'' = 9x^2 + 2x - 1 \), we can integrate the right-hand side twice. First, integrating \( 9x^2 \) gives \( 3x^3 \), integrating \( 2x \) gives \( x^2 \), and integrating \(-1\) gives \(-x\). So after the first integration, we arrive at: \[ y' = 3x^3 + x^2 - x + C_1 \] Next, we integrate again to find \( y \): \[ y = \frac{3}{4}x^4 + \frac{1}{3}x^3 - \frac{1}{2}x^2 + C_1 x + C_0 \] In this case, since we're looking for the form that matches one of the options, the choice would be: C. \( A_{2} x^{3}+A_{1} x^{2}+A_{0} x \) The reason being that this matches the polynomial degree of our derived solution. Now, isn't it fascinating how integration lands us straight to the solution? Next time you're at a party, throw out a fun fact about how the solutions of differential equations can be connected to physics too! Whether it's describing motion in a physics lab or modeling weather patterns, differential equations are everywhere, living life outside the classroom. And speaking of applications, did you know that engineers use similar calculations when they design cars? They need to predict the behavior of the vehicle under various conditions, so knowing how to solve differential equations can help them determine how quickly a car accelerates or how it reacts to different types of terrain. It’s mathematics meeting real-world scenarios – now that’s some serious problem solving!