AHL 9.14 Euler's method for second order equations · AI HL · AHL 9: Differential Equations

AHL 9.14

Euler's method for second order equations

AHL 9.14

Euler's method for second order equations

By the end of this section

Key points

From one second order equation to two first order equations

A second order equation has \(\frac{d^2x}{dt^2}\) in it. Since \(\frac{d^2x}{dt^2} = \frac{d}{dt}\left(\frac{dx}{dt}\right)\), give the first derivative its own letter: \(y = \frac{dx}{dt}\).
Then \(\frac{dy}{dt} = \frac{d^2x}{dt^2}\). If \(\frac{d^2x}{dt^2} = f\left(x, \frac{dx}{dt}, t\right)\), the pair is \(\frac{dx}{dt} = y\) and \(\frac{dy}{dt} = f(x, y, t)\).
Move every term except the second derivative to the right side first. Terms move with their sign changed: \(\frac{d^2x}{dt^2} + 6\frac{dx}{dt} + 4x = 7\) gives \(\frac{dy}{dt} = 7 - 6y - 4x\).
The starting values are \(x(0)\) and \(y(0) = x'(0)\). The second one is the speed, not a position.

Euler's method on the pair

Write the recurrences: \(t_{n+1} = t_n + h\), \(x_{n+1} = x_n + h\,y_n\), \(y_{n+1} = y_n + h\,f(x_n, y_n, t_n)\).
Both new values use the old row. Never use \(y_{n+1}\) to work out \(x_{n+1}\).
If the right side has t in it (such as \(2\cos t\)), use the \(t_n\) of that row, and work in radians.
x at time T is the x of row \(n = T/h\). With \(h = 0.1\), t = 2 is row 20.
Write down the recurrences and a couple of lines of the table. You do not write every row.

Graphs and step lengths

Graph x against t from the table to see the motion. The maximum displacement is the highest point of the graph.
Give a maximum to 2 significant figures when asked. Read x, not y and not t.
A smaller step length gives a more accurate estimate. To justify it, compare both estimates with a run using a much smaller step.
A step that is too long can give nonsense, such as a decaying motion that swings between positive and negative values.

A pendulum

A pendulum's angle satisfies an equation with \(\sin\theta\). It is not linear, so it has no solution by eigenvalues. Euler's method still works.
Write \(\omega=\dfrac{d\theta}{dt}\). Then \(\dfrac{d\theta}{dt}=\omega\) and \(\dfrac{d\omega}{dt}\) is the rest of the equation.
Use radians on the calculator. For a small angle \(\sin\theta\approx\theta\), which gives a linear equation. For a large angle the two models differ.

On the calculator

Euler's method in a spreadsheet

Put t, x and y in three columns. The first row holds t = 0, x(0) and y(0).
The next row of t adds h. The next row of x is the x above plus h times the y above. The next row of y is the y above plus h times the right side of dy/dt, using the x, y and t above.
Fill down until t reaches the time you want, then read the x on that row.
Check that the right side uses the row above, not the row it is in.

Screenshot and video to come

Graphing x against t and finding the maximum

Make a scatter plot of the x column against the t column.
Find the highest point on the graph, or scan down the x column for the largest value.
Read the x value of that point and round it to 2 significant figures.

Screenshot and video to come

Worked examples

Problems like the ones in class. Pick one, try it, then open the working.

Writing a second order equation as a pair

A displacement x satisfies \(\frac{d^2x}{dt^2} + 6\frac{dx}{dt} + 4x = 7\), with x(0) = 3 and x′(0) = −1. Write it as a pair of coupled first order equations and state the starting values.

Working

Name the first derivative
\(y = \frac{dx}{dt}\), so \(\frac{dx}{dt} = y\)

The first equation is always this one.
Isolate the second derivative
\(\frac{d^2x}{dt^2} = 7 - 6\frac{dx}{dt} - 4x\)

Move 6x′ and 4x across: both change sign.
Replace the derivatives
\(\frac{dy}{dt} = 7 - 6y - 4x\)

\(\frac{d^2x}{dt^2}\) is \(\frac{dy}{dt}\) and \(\frac{dx}{dt}\) is \(y\).
Starting values
\(x(0) = 3,\quad y(0) = x'(0) = -1\)

y starts at the starting speed.

Euler's method for a damped motion

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

Working

The pair
\(\frac{dx}{dt} = y,\quad \frac{dy}{dt} = -2y - 3x\)

Let y = dx/dt.
The recurrences
\(x_{n+1} = x_n + 0.1y_n,\quad y_{n+1} = y_n + 0.1(-2y_n - 3x_n)\)

Start from x_0 = 4 and y_0 = −1.
First rows
n = 0: x = 4, y = −1, n = 1: x = 3.9, y = −2, n = 2: x = 3.7, y = −2.77

Row 1: x_1 = 4 + 0.1(−1) = 3.9 and y_1 = −1 + 0.1(−2(−1) − 3(4)) = −2.
From the calculator
x_10 = 0.854745

t = 1 is 10 steps of 0.1.
Answer
\(x(1) \approx 0.855\)

Three significant figures.

A right side that depends on t

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

Working

The pair
\(\frac{dx}{dt} = y,\quad \frac{dy}{dt} = -3y - 4x + 2\cos t\)

The forcing term goes straight into dy/dt.
The recurrences
\(t_{n+1} = t_n + 0.1,\quad x_{n+1} = x_n + 0.1y_n,\quad y_{n+1} = y_n + 0.1(-3y_n - 4x_n + 2\cos t_n)\)

Each row uses its own t_n.
First rows
n = 0: x = 1, y = 0, n = 1: x = 1, y = −0.2, n = 2: x = 0.98, y = −0.341

Row 1: y_1 = 0 + 0.1(2 − 0 − 4) = −0.2.
From the calculator
x_20 = 0.143747

t = 2 is row 20. Radian mode.
Answer
\(x(2) \approx 0.144\)

Reading the maximum from a graph

The displacement x cm of a damped spring t seconds after release satisfies \(\frac{d^2x}{dt^2} + 1.2\frac{dx}{dt} + 9x = 0\), with x(0) = 0 and x′(0) = 6. Use Euler's method with step length 0.1 to graph x for 0 ≤ t ≤ 3, then estimate the maximum displacement to 2 significant figures.

Working

The pair
\(\frac{dx}{dt} = y,\quad \frac{dy}{dt} = -1.2y - 9x\)

Starting values x_0 = 0 and y_0 = 6.
Table of values
n = 0: x = 0, y = 6, n = 1: x = 0.6, y = 5.28, n = 2: x = 1.13, y = 4.11, …

Run the recurrences to t = 3 (30 steps).
Graph x against t
highest point at t = 0.50, x = 1.889

The graph rises, peaks, then falls through zero.
Answer
maximum displacement ≈ 1.9 cm

2 significant figures. The peak is the largest x in the table.

Comparing two step lengths

A motion satisfies \(\frac{d^2x}{dt^2} + 3\frac{dx}{dt} + 2x = 0\), with x(0) = 1 and x′(0) = 0. Use Euler's method to estimate x(4) with step length (a) 1 and (b) 0.5. Which estimate is more accurate?

Working

Step length 1
4 steps: x_4 = −1.00

Rows: x = 1, 1, −1, 1, −1.
Step length 0.5
8 steps: x_8 = 0.00781

Same recurrences with h = 0.5.
A check run
step length 0.1: x(4) ≈ 0.0294

A much smaller step is a good reference.
Compare
|0.00781 − 0.0294| is much smaller than |−1.00 − 0.0294|

The step length 0.5 estimate is closer, so it is more accurate. A smaller step follows the curve more closely. Step length 1 even gives a negative x.

A damped pendulum

A pendulum swings through the air. The angle \(\theta\) radians that it makes with the vertical after \(t\) seconds satisfies \(\dfrac{d^2\theta}{dt^2}+0.4\dfrac{d\theta}{dt}+20\sin\theta=0\), with \(\theta(0)=0.5\) and \(\theta'(0)=0\). Write the equation as a pair of first order equations and use Euler's method with step length 0.1 to estimate \(\theta(1)\).

Working

The pair
\(\dfrac{d\theta}{dt}=\omega,\quad \dfrac{d\omega}{dt}=−0.4\omega-20\sin\theta\)

Let ω = dθ/dt. The sine stays: the equation is not linear.
The recurrences
\(\theta_{n+1}=\theta_n+0.1\omega_n,\quad \omega_{n+1}=\omega_n+0.1(−0.4\omega_n-20\sin\theta_n)\)

Start from θ₀ = 0.5 and ω₀ = 0. Radians.
First rows
n = 0: θ = 0.5, ω = 0; n = 1: θ = 0.500, ω = −0.959; n = 2: θ = 0.404, ω = −1.88

Row 1: ω₁ = 0 + 0.1(0 − 20 sin 0.5) = −0.959.
From the calculator
θ₁₀ = −0.576770

t = 1 is 10 steps of 0.1.
Answer
\(\theta(1)\approx −0.577\) radians

Three significant figures.

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Differential Equations

Euler's method for second order equations

By the end of this section

Key points

From one second order equation to two first order equations

Euler's method on the pair

Graphs and step lengths

A pendulum

On the calculator

Euler's method in a spreadsheet

Graphing x against t and finding the maximum

Worked examples

Writing a second order equation as a pair

Working

Euler's method for a damped motion

Working

A right side that depends on t

Working

Reading the maximum from a graph

Working

Comparing two step lengths

Working

A damped pendulum

Working

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