Mixed practice: Euler's method for second order equations Problem set B Answers

1. (a) \(\frac{dx}{dt} = y,\quad \frac{dy}{dt} = -5y - 9x\)
2. (b) \(\frac{dx}{dt} = y,\quad \frac{dy}{dt} = 4x + 5y - 3\)

1. (a) \(x_{n+1} = x_n + 0.25y_n,\quad y_{n+1} = y_n + 0.25(-4y_n - 7x_n)\)
2. (b) x(0.5) ≈ 1.0625 (and y = −2.625)

x(1.5) ≈ −0.364

x(1.5) ≈ 0.230

1. (a) \(\frac{dx}{dt} = y,\quad \frac{dy}{dt} = -2y - 12x\)
2. (b) maximum ≈ 1.8 cm (at about t = 0.40)

1. (a) \(\frac{dx}{dt} = y,\quad \frac{dy}{dt} = -0.8y - 5x + 2\cos 2t\)
2. (b) x(2.5) ≈ −0.894

1. (a) (i) −0.750, (ii) 0.400
2. (b) Step length 0.5. A run with step length 0.1 gives 0.514, and the step length 0.5 estimate is much closer to it than the step length 1 estimate is

1. (a) \(\dfrac{d\theta}{dt}=\omega,\quad \dfrac{d\omega}{dt}=−0.5\omega−16\sin\theta\)
2. (b) \(\theta(1)\approx−0.539\)
3. (c) \(\theta(1)\approx−0.524\); difference 0.015

1. (a) (i) −64.0, (ii) 0.000488, (iii) 0.00301
2. (b) Step length 1 gives a negative x, which the motion never has. Trust step length 0.25: the smallest step