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


a. Find the approximations to y(0.2) and y(0.4) using Euler’s method with time steps of Δt = 0.2, 0.1, 0.05, and 0.025.


y′(t) = y/2, y(0) = 2; y(t) = 2eᵗᐟ²

Verified step by step guidance

1. Identify the differential equation and initial condition: \(y'(t) = \frac{y}{2}\) with \(y(0) = 2\). The exact solution is given as \(y(t) = 2e^{t/2}\), which will help us check the accuracy of Euler's method approximations.
2. Recall Euler's method formula for approximating the solution at discrete points: \(y_{n+1} = y_n + \Delta t \cdot f(t_n, y_n)\), where \(f(t, y) = \frac{y}{2}\) in this problem.
3. Set up the iterative process starting from \(t_0 = 0\) and \(y_0 = 2\). For each time step size \(\Delta t = 0.2, 0.1, 0.05, 0.025\), compute successive approximations \(y_1, y_2, \ldots\) until you reach \(t = 0.2\) and \(t = 0.4\).
4. For example, with \(\Delta t = 0.2\), calculate \(y_1\) at \(t_1 = 0.2\) using \(y_1 = y_0 + 0.2 \times \frac{y_0}{2}\), then calculate \(y_2\) at \(t_2 = 0.4\) similarly using \(y_1\). Repeat this process for the smaller step sizes, which will require more iterations.
5. Compare the approximations obtained for each \(\Delta t\) to observe how the step size affects the accuracy of Euler's method. Smaller \(\Delta t\) values generally yield better approximations to the exact solution \(y(t) = 2e^{t/2}\).

Key Concepts

Euler's Method

Euler's method is a numerical technique to approximate solutions of first-order differential equations using discrete steps. Starting from an initial value, it estimates the next value by moving along the slope given by the differential equation multiplied by the step size. Smaller step sizes generally improve accuracy but require more computations.

Initial Value Problems (IVPs)

An initial value problem specifies a differential equation along with a starting point (initial condition) for the solution. The goal is to find the function that satisfies both the differential equation and the initial condition, which serves as the basis for numerical methods like Euler's method.

Error and Step Size in Numerical Methods

The error in Euler's method depends on the step size Δt; smaller steps reduce the local truncation error, leading to more accurate approximations. Understanding how step size affects error helps in balancing computational effort and solution precision when approximating values like y(0.2) and y(0.4).