Mixed practice: second order equations, exact solutions and models Problem set B Answers

Substitution gives 0 = 0, and x(0) = 3, x′(0) = 1

1. (a) \(\frac{dx}{dt} = y,\quad \frac{dy}{dt} = -5y - 4x;\quad \begin{pmatrix} 0 & 1 \\ -4 & -5 \end{pmatrix}\)
2. (b) \(\lambda = -1: \begin{pmatrix} 1 \\ -1 \end{pmatrix};\quad \lambda = -4: \begin{pmatrix} 1 \\ -4 \end{pmatrix}\)
3. (c) \(x = 4e^{-t} + e^{-4t}\)
4. (d) x(1) = 1.49

1. (a) 0.188 (h = 1) and 0.350 (h = 0.5)
2. (b) x(4) = 0.472
3. (c) 60.3% (h = 1) and 25.8% (h = 0.5)

1. (a) Oscillation starting at 0, with peaks at t = 0.7 and t = 3.3
2. (b) amplitude ≈ 1.2, period ≈ 2.6

1. (a) 1.5 mm (at t = 7.9)
2. (b) 5.8 mm (at t = 7.4)
3. (c) ω = 2.8: it is closer to the natural frequency

1. (a) n falls from 1.5 to a minimum of −2.6 at t = 0.8, then oscillates about a level near 0.8 with swings that slowly get smaller
2. (b) n goes below 0, but a concentration cannot be negative

1. (a) t = 1.3
2. (b) 1.6 years
3. (c) 2.7 dollars
4. (d) 3.5 dollars
5. (e) The price follows a smooth wave to a fixed level; real prices respond to news and chance

1. (a) x(1) ≈ 1.81, x(2) ≈ 7.64
2. (b) Left side equals right side, and x(0) = 0, x′(0) = 0
3. (c) 9.40% and 4.56%