Final answer:
The particular solution to the differential equation y'' - y = 4e^(2t) is found using the method of undetermined coefficients. By guessing a solution of the form Ae^(2t) and equating coefficients, the particular solution is determined to be y_p = (4/3)e^(2t).
Step-by-step explanation:
The student is asking for help in finding a particular solution to the second-order non-homogeneous differential equation y'' - y = 4e^(2t). To find a particular solution, we can use the method of undetermined coefficients. Since the right-hand side is an exponential function, we guess a particular solution of the form Ae^(2t), where A is a constant to be determined.
We then differentiate this guess twice to get y'' = 4Ae^(2t), and substitute y and y'' into the differential equation to find A. Doing this, we have 4Ae^(2t) - Ae^(2t) = 4e^(2t), which simplifies to 3Ae^(2t) = 4e^(2t). By equating coefficients, we find that A = 4/3. The particular solution is therefore y_p = (4/3)e^(2t).
This method is directly applicable to similar problems involving linear differential equations with constant coefficients.