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


b. Using the exact solution given, compute the errors in the Euler approximations at t=0.2 and t=0.4.


y′(t) = −y, y(0) = 1; y(t) = e⁻ᵗ

Verified step by step guidance

1. Identify the given initial value problem (IVP): \(y'(t) = -y\), with initial condition \(y(0) = 1\), and the exact solution \(y(t) = e^{-t}\).
2. Recall that Euler's method approximates the solution using the formula: \(y_{n+1} = y_n + h f(t_n, y_n)\), where \(h\) is the step size and \(f(t, y) = y'(t)\).
3. Choose the step size \(h\) based on the points where the approximation is needed. Since we want approximations at \(t=0.2\) and \(t=0.4\), a natural choice is \(h=0.2\).
4. Compute the Euler approximations step-by-step: starting from \(t_0=0\), \(y_0=1\), calculate \(y_1\) at \(t_1=0.2\) and then \(y_2\) at \(t_2=0.4\) using the Euler update formula.
5. Calculate the exact values at \(t=0.2\) and \(t=0.4\) using \(y(t) = e^{-t}\), then find the errors by subtracting the Euler approximations from the exact values: \(\text{Error} = |y_{exact} - y_{Euler}|\) at each point.

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 given by the differential equation at the current point. This method is simple but can accumulate errors depending on the step size.

Exact Solution of Differential Equations

The exact solution is the precise function that satisfies the differential equation and initial conditions. In this problem, y(t) = e⁻ᵗ is the exact solution, which allows comparison with numerical approximations to evaluate their accuracy. Knowing the exact solution helps quantify the error in numerical methods.

Error Analysis in Numerical Methods

Error analysis involves calculating the difference between the exact solution and the numerical approximation at specific points. It helps assess the accuracy and reliability of methods like Euler's. Errors depend on factors like step size and the nature of the differential equation.