Euler's method: accuracy and contexts Problem set B

\(\dfrac{dy}{dx}=6x^2\) and \(y=2\) when \(x=0\).

1. (a) Use Euler's method with \(h=1\) to estimate \(y(2)\) and \(y(4)\).
2. (b) Use Euler's method with \(h=0.5\) to estimate \(y(2)\) and \(y(4)\).
3. (c) Solve the differential equation exactly. Hence say which of your four estimates is furthest from the true value.

\(\dfrac{dy}{dx}=0.25y\) and \(y=8\) when \(x=0\).

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

\(\dfrac{dy}{dx}=\dfrac{x^2}{1+y}\) and \(y=1\) when \(x=1\). Use Euler's method with \(h=0.25\) to estimate \(y(2)\).

\(\dfrac{dy}{dx}=2y^2\) and \(y=0.25\) 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.

\(\dfrac{dy}{dx}=\dfrac{3}{x+y+1}\) and \(y=2\) when \(x=0\). Use Euler's method with \(h=0.5\) and technology to estimate \(y(10)\).

The depth \(d\) cm of snow on a pitch, \(t\) days after the thaw begins, is modelled by \(\dfrac{dd}{dt}=-0.3d-0.2t\). At the start \(d=12\). Use Euler's method with a step length you choose to estimate the time, to the nearest day, when the pitch is clear of snow.

The amounts of solution \(x\) litres and \(y\) litres in two connected tanks, \(t\) hours after a pump starts, are modelled by \(\dfrac{dx}{dt}=0.2y-0.3x\) and \(\dfrac{dy}{dt}=0.3x-0.2y-0.1t\). At the start \(x=50\) and \(y=20\). Use Euler's method with \(h=0.25\) to estimate \(x\) and \(y\) after 1 hour.

A train's speed is measured every 6 seconds as it brakes. The table shows the speeds.

1. (a) Use Euler's method to estimate the distance travelled in the first 30 seconds.
2. (b) Is your estimate too large or too small? Give a reason.

\(\dfrac{dy}{dx}=3\sin x-0.6y\) with \(y=2\) when \(x=0\) (\(x\) in radians).

1. (a) Use Euler's method with \(h=0.1\) and technology to sketch the solution for \(0\le x\le6\).
2. (b) Hence estimate, to 1 decimal place, the largest value of \(y\) in this interval.