Consider the differential equation \(\frac{dy}{dx} = y – x^2 + 1\) with the initial condition \(y(0) = 0.5\).
(a) Apply Euler’s method with a step size of \(h = 0.2\) to approximate \(y(0.4)\).
(b) Given that the exact solution at \(x = 0.4\) is \(y(0.4) = 1.433\), discuss the potential error due to the chosen step size by comparing the Euler approximation with the exact value. Explain how graphing software or spreadsheets can be used to visually explore and understand the nature of this error as the step size changes.
Question
Apply Euler’s method with step size \(h = 0.2\) to approximate \(y(0.4)\) for the differential equation \(\frac{dy}{dx} = y – x^2 + 1\) with \(y(0)=0.5\). Show each step of your calculation.
Hint
Hint
Hint
Remember Euler’s formula: \(y_{n+1} = y_n + h\,f(x_n, y_n)\). Identify \(f(x,y)\) from the given differential equation and perform iterative calculations.
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Answer
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- Formula Booklet
Complete Answer
Approximately \(y(0.4) \approx 1.152\)
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Question
Discuss the potential error of the Euler approximation \(y(0.4) \approx 1.152\) compared to the exact value \(y(0.4)=1.433\). Explain how technology, such as graphing software or spreadsheets, can be used to explore how the error changes with different step sizes over the interval \([0, 0.4]\).
Hint
Hint
Hint
Calculate the absolute error. Consider how plotting both the approximate and exact solutions using technology can visually represent the error. Think about the effect of reducing the step size \(h\) and observe changes using technological tools.
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- Formula Booklet
Complete Answer
The Euler method underestimates the exact value by approximately \(0.281\). Technology can be used to plot both the Euler approximation and the exact solution, visually demonstrating the divergence and error accumulation. Reducing the step size in technological tools can demonstrate improved accuracy and error reduction, which can be observed both graphically and numerically.
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