Chapter 3, Problem 3.22P

Let the initial position and speed of an overdamped, nondriven oscillator be x₀ and v₀, respectively.

1. (a) Show that the values of the amplitudes A₁ and A₂ in Equation 3.44 have the values $A_1\text{=}\frac{{\beta }_2{x}_0+v_0}{{\beta }_2-{\beta }_1}$ and $A_2\text{=}-\frac{{\beta }_1{x}_0+v_0}{{\beta }_2-{\beta }_1}$ where β₁ = β − ω₂ and β₂ = β + ω₂.
2. (b) Show that when A₁ = 0, the phase paths of Figure 3-11 must be along the dashed curve given by $\stackrel{\dot{}}{x}\text{=}-{\beta }_{2}x$, otherwise the asymptotic paths are along the other dashed curve given by $\stackrel{\dot{}}{x}\text{=}-{\beta }_{1}x$. Hint: Note that β₂ > β₁ and find the asymptotic paths when t → ∞.