Final Answer:
The solution to the differential equation y" + 9y' = 972 sin(9t) + 1296 cos(9t) with initial conditions y(0) = 4 and y'(0) = 2 is y = 81 sin(9t) + 4 cos(9t) + 3.
Step-by-step explanation:
To solve the differential equation y" + 9y' = 972 sin(9t) + 1296 cos(9t), first, we find the complementary solution by solving the homogeneous equation y" + 9y' = 0. The characteristic equation is m^2 + 9m = 0, which gives the characteristic roots m = 0 and m = -9. Therefore, the complementary solution is y_c = c1 + c2 * e^(-9t), where c1 and c2 are constants.
Next, to find a particular solution for the non-homogeneous equation, we use the method of undetermined coefficients. Since the right-hand side contains sinusoidal functions, we guess a particular solution in the form of y_p = A sin(9t) + B cos(9t). Taking the derivatives and substituting into the differential equation gives -81A sin(9t) + 81B cos(9t) + 81A cos(9t) + 81B sin(9t) = 972 sin(9t) + 1296 cos(9t). Equating coefficients of sin(9t) and cos(9t) separately, we get A = 81 and B = 4. Therefore, the particular solution is y_p = 81 sin(9t) + 4 cos(9t).
Combining the complementary and particular solutions gives the general solution y = y_c + y_p = c1 + c2 * e^(-9t) + 81 sin(9t) + 4 cos(9t). Using the initial conditions y(0) = 4 and y'(0) = 2, we find c1 = 3 and c2 = 0. Hence, the solution to the differential equation with the given initial conditions is y = 81 sin(9t) + 4 cos(9t) + 3.