Final answer:
The solution involves finding the complementary solution of the homogeneous equation, then determining a particular solution using the method of undetermined coefficients, after which we combine both and apply the initial conditions to find the constants.
Step-by-step explanation:
The student is asking to find the solution to the second order linear differential equation with constant coefficients y'' + 2y' + y = 96eu, subject to the initial conditions y(0)=3 and y'(0)=7. To solve this, we first find the complementary solution by solving the homogeneous equation y'' + 2y' + y = 0, and then find a particular solution to the non-homogeneous equation by using the method of undetermined coefficients.
For the homogeneous equation, we look for solutions of the form y = ert. Substituting into the homogeneous equation gives us a characteristic equation r2 + 2r + 1 = 0, which factors to (r + 1)2 = 0. So we have a repeated root r = -1, and the complementary solution is yc = (C1 + C2t)e-t, where C1 and C2 are constants to be determined.
To find the particular solution yp, we guess a solution of the form yp = Aeu, where A is an undetermined coefficient. We substitute yp into the non-homogeneous equation and solve for A. Combining the complementary and particular solutions gives the general solution y = yc + yp. Finally, we apply the initial conditions to find the constants C1 and C2.