Final answer:
The general solution for x in terms of t is x(t) = C₁e^(2t) + C₂e^(-3t) + (160/13)cos(4t) + (53/13)sin(4t), where C₁ and C₂ are constants determined by the initial conditions.
Step-by-step explanation:
The given differential equation is d²x/dt² = αdx/dt - kx + F(x,t), where α = 6 is the drag coefficient, k = 12 is the Hooke's constant, and F(x,t) = 160cos(4t) - x is an additional external force. To find the general solution for x in terms of t, we need to solve this differential equation.
We can start by assuming a solution of the form x(t) = Ae^(λt), where A is a constant and λ is an unknown parameter. By substituting this solution into the differential equation, we can determine the values of λ that satisfy the equation and find the general solution.
After solving the differential equation, we find that the general solution for x in terms of t is x(t) = C₁e^(2t) + C₂e^(-3t) + (160/13)cos(4t) + (53/13)sin(4t), where C₁ and C₂ are constants determined by the initial conditions.