29–32. {Use of Tech} Errors in Euler’s method Consider the following initial value problems.


c. Which time step results in the more accurate approximation? Explain your observations.


y′(t) = 4−y, y(0) = 3; y(t) = 4−e⁻ᵗ

Verified step by step guidance

1. Identify the given initial value problem (IVP): \(y'(t) = 4 - y\), with initial condition \(y(0) = 3\), and the exact solution \(y(t) = 4 - e^{-t}\).
2. Recall that Euler's method approximates the solution by using the formula \(y_{n+1} = y_n + h f(t_n, y_n)\), where \(h\) is the time step size and \(f(t, y) = 4 - y\) in this case.
3. Understand that the accuracy of Euler's method depends on the size of the time step \(h\): smaller \(h\) generally leads to a more accurate approximation because it reduces the local truncation error at each step.
4. To compare the accuracy for different time steps, compute the approximate values of \(y\) at a fixed time using Euler's method with each time step, then compare these approximations to the exact solution \(y(t) = 4 - e^{-t}\) at the same time.
5. Observe that the time step with the smaller \(h\) will produce values closer to the exact solution, demonstrating that a smaller time step results in a more accurate approximation in Euler's method.

Key Concepts

Euler's Method

Euler's method is a numerical technique to approximate solutions of initial value problems for differential equations. It uses a stepwise approach, estimating the next value by moving along the slope (derivative) at the current point. The accuracy depends on the size of the time step chosen.

Time Step Size and Accuracy

The time step size in Euler's method determines the intervals at which the solution is approximated. Smaller time steps generally yield more accurate results because they better capture the curve's behavior, while larger steps can introduce significant errors and reduce precision.

Exact Solution and Error Analysis

Comparing the numerical approximation to the exact solution allows for error analysis. The exact solution y(t) = 4 - e^{-t} provides a benchmark to measure how close Euler's method approximations are, helping to understand how step size affects the error magnitude.