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

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

1. (a) \(\frac{dx}{dt} = y,\quad \frac{dy}{dt} = -3y + 4x;\quad \begin{pmatrix} 0 & 1 \\ 4 & -3 \end{pmatrix}\)
2. (b) \(\lambda = 1: \begin{pmatrix} 1 \\ 1 \end{pmatrix};\quad \lambda = -4: \begin{pmatrix} 1 \\ -4 \end{pmatrix}\)
3. (c) \(x = 2e^{t} + e^{-4t}\)
4. (d) x(1) = 5.45

1. (a) 0.500 (h = 1) and 0.681 (h = 0.5)
2. (b) x(3) = 0.793
3. (c) 36.9% (h = 1) and 14.2% (h = 0.5)

1. (a) Oscillation starting at 0, with peaks at t = 0.6 and t = 2.7
2. (b) amplitude ≈ 1.3, period ≈ 2.1

1. (a) 0.83 cm (at t = 7.8)
2. (b) 3.8 cm (at t = 7.2)
3. (c) ω = 2.2: it is closer to the natural frequency

1. (a) n falls from 2 to a minimum of −3.0 at t = 0.6, then oscillates about a level near 0.7 with swings that slowly get smaller
2. (b) n goes below 0, but the number of bacteria cannot be negative

1. (a) t = 1.6
2. (b) 1.8 years
3. (c) 3.2 dollars
4. (d) 4.0 dollars
5. (e) The price follows a smooth wave to a fixed level; real prices respond to news and chance

1. (a) x(1) ≈ 0.908, x(2) ≈ 3.83
2. (b) Left side equals right side, and x(0) = 0, x′(0) = 0
3. (c) 9.19% and 4.25%