Euler’s Method Iteration Table Construction

00:00 12 marks

Use Euler’s method to approximate the solution of the second order differential equation \(y” = -0.5y\) with initial conditions \(y(0)=2\) and \(y'(0)=0\) over the interval \(0 \le x \le 0.4\) using a step size of \(h = 0.2\).

(a) Construct an iteration table showing the approximate values of \(y\) and \(y’\) at each step.

(b) Compare the approximate value of \(y(0.4)\) with the expected behavior of the solution.

Part a

6 minutes 6 marks

Question

Construct an iteration table showing the values of \(y\) and \(y’\) at \(x = 0, 0.2, 0.4\) using Euler’s method for the system rewritten as \(y’ = v\) and \(v’ = -0.5y\).

Hint Hint

Hint

Rewrite as first order system: \(y’ = v\) and \(v’ = -0.5y\). Then update with \(y_{n+1} = y_n + h \cdot v_n\) and \(v_{n+1} = v_n + h \cdot (-0.5y_n)\).

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Complete Answer

x	y	y’
0	2	0
0.2	2	-0.2
0.4	1.96	-0.4

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Part b

6 minutes 6 marks

Question

Compare the approximate value of \(y(0.4)\) obtained from Euler’s method with the expected behavior of the solution for \(y” = -0.5y\).

Hint Hint

Hint

Consider the effect of the negative coefficient in \(y” = -0.5y\) on the trend of \(y\), and discuss if the decrease from \(y(0)=2\) is reasonable.

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Complete Answer

The approximate value \(y(0.4) \approx 1.96\) shows a slight decrease from the initial value, which is consistent with the expected behavior of the system where the acceleration is proportional to \(-0.5y\), indicating a restoring force leading to oscillatory or decaying behavior.

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