Final answer:
To solve the differential equation y'' - 3y' + 2y = e^t*sin(t), determine the homogeneous solution first and then find a particular solution that incorporates the e^t*sin(t) term. Combine these solutions to obtain the general solution.
Step-by-step explanation:
Finding a general solution to the given differential equation involves two parts: solving the homogeneous equation and finding a particular solution for the nonhomogeneous part.
Homogeneous Solution
First, we solve the homogeneous equation y'' - 3y' + 2y = 0. This characteristic equation is r^2 - 3r + 2 = 0 which factors to (r - 2)(r - 1) = 0. Hence the solutions for r are r = 1 and r = 2. Using these roots, the general solution of the homogeneous equation is Yh = C1e^t + C2e^(2t), where C1 and C2 are constants.
Particular Solution
To find a particular solution Yp to the nonhomogeneous equation, we can try a method such as undetermined coefficients or variation of parameters. In this case, because the right-hand side of the equation is e^t*sin(t), we might assume a solution of the form Yp = t(Ae^t*cos(t) + Be^t*sin(t)) to account for the resonance with the e^t term in the homogeneous solution.
We differentiate this assumed Yp to find Yp' and Yp'', then plug these into the original equation to determine the coefficients A and B.
Total Solution
Combining the homogeneous and particular solutions, the general solution to the original differential equation is Y = Yh + Yp.