Final answer:
The general solution of the differential equation y" - 2y' + y = e^t/(1 + t^2) can be found using the method of undetermined coefficients. The complementary solution is y_c = C_1e^t + C_2te^t and the particular solution is y_p = (1/2)t^2e^t. The general solution is y = y_c + y_p = C_1e^t + C_2te^t + (1/2)t^2e^t.
Step-by-step explanation:
The general solution of the differential equation y" - 2y' + y = e^t/(1 + t^2) can be found using the method of undetermined coefficients. First, we find the complementary solution by solving the homogeneous equation: y" - 2y' + y = 0. The characteristic equation is r^2 - 2r + 1 = 0, which has a repeated root of r = 1. Therefore, the complementary solution is y_c = C_1e^t + C_2te^t, where C_1 and C_2 are constants.
To find the particular solution, we assume that it has the form y_p = At^2e^t. Plugging this into the differential equation, we find that A = 1/2. Therefore, the particular solution is y_p = (1/2)t^2e^t.
Finally, the general solution is the sum of the complementary and particular solutions: y = y_c + y_p = C_1e^t + C_2te^t + (1/2)t^2e^t, where C_1 and C_2 are constants.