Final answer:
To solve the second-order equation describing a damped harmonic oscillator with a forcing function, we first find the homogeneous solution and the particular solution. The homogeneous solution is found by setting the forcing function to zero, and the particular solution is found using the method of undetermined coefficients. The general solution is then the sum of the homogeneous solution and the particular solution.
Step-by-step explanation:
To solve the second-order equation describing a damped harmonic oscillator with a forcing function, we first need to find the homogeneous solution and the particular solution.
The homogeneous solution is found by setting the forcing function to zero: y'' + 9y' + 8y = 0. This is a characteristic equation with solutions of the form y = e^(rt). Plugging this into the equation, we get r^2 + 9r + 8 = 0. Factoring this quadratic equation, we get (r + 8)(r + 1) = 0. Therefore, the two homogeneous solutions are y1 = e^(-8t) and y2 = e^(-t).
The particular solution can be found by using the method of undetermined coefficients. Since the forcing function is of the form Acos(t) - Bsin(t), we assume a particular solution of the form y_p = Ccos(t) + Dsin(t). Plugging this into the equation, we find that C = -7/8 and D = 9/8. Therefore, the particular solution is y_p = (-7/8)cos(t) + (9/8)sin(t).
Finally, the general solution is given by the sum of the homogeneous solution and the particular solution: y = c1e^(-8t) + c2e^(-t) + (-7/8)cos(t) + (9/8)sin(t), where c1 and c2 are constants determined by the initial conditions.