Question

ForCED Damped Harmonic Motion In the physical world damping is always present, thus we should consider what happens when we add some damping to our harmonic oscillator model. This is done by adding a term cx′ where c is a constant, x′′ +cx′ +ω₀² x=Acos(ωt) Consider the nonhomogenous differential initial value problem 0.2x′′ +1.2x′ +2x=5cos(4t)x(0)=0.5,x′ (0)=0. If the above IVP is solved, then the complete set of terms of which the solution is comprised are listed below: (a) −25/102 cos(4t) (b) 50/51 sin(4t) (c) −e⁻³ᵗ86/51 sin(t) (d) e⁻³ᵗ38/51cos(t)

Answer 1

The correct set of terms for the solution of the given forced damped harmonic motion initial value problem is (c) −e⁻³ᵗ86/51 sin(t).

Step-by-step explanation:

In the provided differential equation 0.2x′′ +1.2x′ +2x = 5cos(4t), the terms involve damping (c) and a forcing term (5cos(4t)). To find the particular solution, we assume a solution of the form x_p = Acos(4t) + Bsin(4t) and substitute it into the differential equation.

After calculating the derivatives and substituting back into the equation, we find that the particular solution is x_p = −e⁻³ᵗ86/51 sin(t). This term accounts for the effects of damping and the external force on the system.

The initial conditions x(0) = 0.5 and x′(0) = 0 are then applied to determine the values of the constants, resulting in the specific solution for the forced damped harmonic motion problem. The correct term, therefore, is the one that satisfies both the differential equation and the initial conditions, which is −e⁻³ᵗ86/51 sin(t).