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

Write each equation as a pair of coupled first order equations, using y = dx/dt.

1. (a) \(\frac{d^2x}{dt^2} + 7\frac{dx}{dt} + 2x = 0\)
2. (b) \(\frac{d^2x}{dt^2} = 3x - 2\frac{dx}{dt} + 4\)

A motion satisfies \(\frac{d^2x}{dt^2} + 3\frac{dx}{dt} + 5x = 0\), with x(0) = 2 and x′(0) = −3. Use Euler's method with step length 0.2.

1. (a) Write down the recurrences for \(x_{n+1}\) and \(y_{n+1}\).
2. (b) Work out x and y after two steps, and so estimate x(0.4).

Given \(\frac{d^2x}{dt^2} + 2\frac{dx}{dt} + 8x = 0\), x(0) = 3 and x′(0) = 0, use Euler's method with step length 0.1 and the calculator to estimate x(2).

A function satisfies \(\frac{d^2x}{dt^2} + 2\frac{dx}{dt} + 5x = 5t\), with x(0) = 0 and x′(0) = 1. Use Euler's method with step length 0.1 to estimate x(1.5).

The displacement x mm of the arm of a vibration sensor, t seconds after it is knocked, satisfies \(\frac{d^2x}{dt^2} + 1.5\frac{dx}{dt} + 7x = 0\), with x(0) = 0 and x′(0) = 5.

1. (a) Write the equation as a pair of coupled first order equations.
2. (b) Use Euler's method with step length 0.1 and technology to graph x for 0 ≤ t ≤ 4. Estimate the maximum displacement to 2 significant figures.

The sideways displacement x cm of a model footbridge deck, t seconds after wind gusts begin, satisfies \(\frac{d^2x}{dt^2} + 0.5\frac{dx}{dt} + 4x = 3\sin t\), with x(0) = 0 and x′(0) = 0.

1. (a) Write the equation as a pair of coupled first order equations.
2. (b) Use Euler's method with step length 0.1 to estimate x(3).

A motion satisfies \(\frac{d^2x}{dt^2} + 4\frac{dx}{dt} + 3x = 0\), with x(0) = 1 and x′(0) = 0.

1. (a) Use Euler's method to estimate x(4) with step length (i) 1 and (ii) 0.5.
2. (b) Which estimate is more accurate? Justify your answer.

A child on a swing is pulled back and released. The angle \(\theta\) radians that the swing makes with the vertical after \(t\) seconds satisfies \(\dfrac{d^2\theta}{dt^2}+0.3\dfrac{d\theta}{dt}+9\sin\theta=0\), with \(\theta(0)=0.6\) and \(\theta'(0)=0\).

1. (a) Write the equation as a pair of coupled first order equations.
2. (b) Use Euler's method with step length 0.1 to estimate \(\theta(1)\), to 3 significant figures.
3. (c) Replace \(\sin\theta\) by \(\theta\) and repeat the estimate. Find the difference between the two estimates, to 2 significant figures.

The height x cm of a door-closer piston above its rest position, t seconds after release, satisfies \(\frac{d^2x}{dt^2} + 3\frac{dx}{dt} + 2x = 0\), with x(0) = 2 and x′(0) = 1. The motion is heavily damped: x rises a little at first, then falls towards 0 and never becomes negative.

1. (a) Use Euler's method to estimate x(6) with step length (i) 1, (ii) 0.5 and (iii) 0.25.
2. (b) One estimate cannot be right. Say which, give a reason, and say which estimate you trust most.