1. **Problem Statement:** Given the differential equation $ay'' + by' + cy = xe^x$ with the condition $b^2 - 4ac < 0$, find the correct form of the particular solution $y_p$ using the method of undetermined coefficients. 2. **Background:** The characteristic equation associated with the homogeneous part is $ar^2 + br + c = 0$. Since $b^2 - 4ac < 0$, the roots are complex conjugates, so the homogeneous solution $y_h$ involves terms like $e^{ ext{Re}(r)x}$ times sine and cosine. 3. **Method of Undetermined Coefficients:** For a right-hand side of the form $xe^x$, the particular solution guess depends on whether $e^x$ is a solution to the homogeneous equation. 4. **Check if $e^x$ is a solution to the homogeneous equation:** Substitute $r=1$ into the characteristic equation: $$a(1)^2 + b(1) + c = a + b + c$$ If $a + b + c = 0$, then $e^x$ is a solution to the homogeneous equation. 5. **Since $b^2 - 4ac < 0$, roots are complex and not equal to 1, so $e^x$ is not a solution to the homogeneous equation.** 6. **Form of the particular solution:** For $xe^x$ on the right side and $e^x$ not a solution of the homogeneous equation, the particular solution is of the form: $$y_p = (A x + B) e^x$$ 7. **Check the options:** - I. $y_p = Ax + Be^x$ (incorrect, terms not multiplied properly) - II. $y_p = Ax + Bxe^x$ (incorrect, missing $e^x$ in the first term) - III. $y_p = Axe^x + Be^x$ (correct form) - IV. $y_p = Ae^x$ (too simple, does not account for $x$) - V. $y_p = A + Be^x$ (incorrect, constant term without $e^x$) **Final answer:** The particular solution is $y_p = Axe^x + Be^x$ (Option III).