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

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

1. (a) \(\frac{d^2x}{dt^2} + 5\frac{dx}{dt} + 9x = 0\)
2. (b) \(\frac{d^2x}{dt^2} = 4x + 5\frac{dx}{dt} - 3\)

A motion satisfies \(\frac{d^2x}{dt^2} + 4\frac{dx}{dt} + 7x = 0\), with x(0) = 1 and x′(0) = 2. Use Euler's method with step length 0.25.

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(0.5).

Given \(\frac{d^2x}{dt^2} + 3\frac{dx}{dt} + 10x = 0\), x(0) = 1 and x′(0) = 4, use Euler's method with step length 0.1 and the calculator to estimate x(1.5).

A function satisfies \(\frac{d^2x}{dt^2} + 4\frac{dx}{dt} + 6x = 3e^{-t}\), with x(0) = 1 and x′(0) = 0. Use Euler's method with step length 0.1 to estimate x(1.5).

The height x cm of a diving board tip above its rest position, t seconds after a jump, satisfies \(\frac{d^2x}{dt^2} + 2\frac{dx}{dt} + 12x = 0\), with x(0) = 0 and x′(0) = 7.

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 height above rest to 2 significant figures.

The displacement x mm of a speaker cone, t milliseconds after a signal begins, satisfies \(\frac{d^2x}{dt^2} + 0.8\frac{dx}{dt} + 5x = 2\cos 2t\), 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(2.5).

A motion satisfies \(\frac{d^2x}{dt^2} + 2.5\frac{dx}{dt} + x = 0\), with x(0) = 3 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 lamp hangs from a long chain in a hall and is nudged. The angle \(\theta\) radians that the chain makes with the vertical after \(t\) seconds satisfies \(\dfrac{d^2\theta}{dt^2}+0.5\dfrac{d\theta}{dt}+16\sin\theta=0\), with \(\theta(0)=0.4\) 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 depth x cm of liquid in a slowly draining tube above its final level, t seconds after the tap opens, satisfies \(\frac{d^2x}{dt^2} + 4\frac{dx}{dt} + 3x = 0\), with x(0) = 2 and x′(0) = 0. The motion is heavily damped: x falls steadily 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.