Final answer:
To solve y" + 2y' + y = x²e⁻¹, find the complementary solution of the associated homogeneous equation, then propose and solve for the particular solution using undetermined coefficients. Combine these to get the final solution.
Step-by-step explanation:
The differential equation provided is y" + 2y' + y = x²e⁻¹. To solve this by the method of undetermined coefficients, first solve the associated homogeneous equation, y" + 2y' + y = 0, to find the complementary solution. The characteristic equation of the homogeneous equation is r² + 2r + 1 = 0, which has a repeated root of r = -1. Thus, the complementary solution is y_c = (A + Bx)e⁻¹, where A and B are constants. Next, for the particular solution, y_p, propose a solution of the form y_p = x²(Ax² + Bx + C)e⁻¹, where A, B, and C are coefficients to be determined. After plugging y_p into the original equation and equating the coefficients, you'll solve for A, B, and C. The final solution is the sum of the complementary and particular solutions, y = y_c + y_p.