Euler's method: accuracy and contexts Problem set A

\(\dfrac{dy}{dx}=3x^2\) and \(y=1\) when \(x=0\).

1. (a) Use Euler's method with \(h=0.5\) to estimate \(y(1)\) and \(y(2)\).
2. (b) Use Euler's method with \(h=0.25\) to estimate \(y(1)\) and \(y(2)\).
3. (c) Solve the differential equation exactly. Hence say which of your four estimates is furthest from the true value.

\(\dfrac{dy}{dx}=-0.5y\) and \(y=6\) when \(x=0\).

1. (a) Use Euler's method with \(h=0.5\) to estimate \(y(2)\).
2. (b) Solve the differential equation exactly and find \(y(2)\).
3. (c) Find the percentage error of your estimate.
4. (d) Explain how the estimate could be made more accurate.

\(\dfrac{dy}{dx}=xy^2\) and \(y=0.4\) when \(x=1\). Use Euler's method with \(h=0.2\) to estimate \(y(2)\).

\(\dfrac{dy}{dx}=y^2\) and \(y=0.5\) when \(x=0\). The graph shows the exact solution and the Euler polyline with \(h=0.25\).

1. (a) Use Euler's method with \(h=0.25\) to estimate \(y(1)\).
2. (b) Solve the differential equation exactly and find \(y(1)\).
3. (c) Use the shape of the exact solution to explain why the Euler estimate is smaller than the true value.

\(\dfrac{dy}{dx}=\dfrac{1}{x+y}\) and \(y=1\) when \(x=0\). Use Euler's method with \(h=0.5\) and technology to estimate \(y(8)\).

During a drought, the volume \(V\) million litres of water in a reservoir, \(t\) weeks after the rains stop, is modelled by \(\dfrac{dV}{dt}=-0.25\sqrt{V}-0.1t\). At the start \(V=12\). Use Euler's method with a step length you choose to estimate the time, to the nearest week, when the reservoir is empty.

The numbers of voles \(x\) and owls \(y\) in a wood, \(t\) years after a survey starts, are modelled by \(\dfrac{dx}{dt}=0.8x-0.02xy\) and \(\dfrac{dy}{dt}=0.005xy-0.4y\). At the start there are 100 voles and 30 owls. Use Euler's method with \(h=0.1\) to estimate \(x\) and \(y\) after 0.5 years.

A sprinter's speed is measured every second after the start of a race. The table shows the speeds.

1. (a) Use Euler's method to estimate the distance travelled in the first 5 seconds.
2. (b) Is your estimate too large or too small? Give a reason.

\(\dfrac{dy}{dx}=2\sin x-0.5y\) with \(y=3\) when \(x=0\) (\(x\) in radians).

1. (a) Use Euler's method with \(h=0.1\) and technology to sketch the solution for \(0\le x\le8\).
2. (b) Hence estimate, to 1 decimal place, the largest value of \(y\) in this interval.