33–36. {Use of Tech} Computing Euler approximations Use a calculator or computer program to carry out the following steps.
a. Approximate the value of y(T) using Euler’s method with the given time step on the interval [0,T].


y′(t) = -2y, y(0) = 1; Δt = 0.2, T = 2; y(t) = e⁻²ᵗ

Verified step by step guidance

1. Identify the differential equation and initial condition: \(y'(t) = -2y\), with \(y(0) = 1\).
2. Set the time step \(\Delta t = 0.2\) and the interval from \(t=0\) to \(T=2\). Calculate the number of steps as \(n = \frac{T}{\Delta t} = \frac{2}{0.2} = 10\).
3. Recall Euler's method formula: \(y_{k+1} = y_k + \Delta t \cdot f(t_k, y_k)\), where \(f(t, y) = y'(t) = -2y\).
4. Start with the initial value \(y_0 = 1\) at \(t_0 = 0\). For each step \(k\) from 0 to 9, compute \(y_{k+1}\) using the formula: \(y_{k+1} = y_k + 0.2 \times (-2 y_k) = y_k - 0.4 y_k = y_k (1 - 0.4)\).
5. Repeat the calculation iteratively for all steps until you reach \(t = 2\). The final value \(y_{10}\) will be the Euler approximation of \(y(2)\).

Key Concepts

Euler's Method

Euler's method is a numerical technique to approximate solutions of first-order differential equations. It uses a stepwise approach, updating the solution by moving along the slope given by the differential equation at each step. This method is especially useful when an exact solution is difficult to find.

Initial Value Problems (IVP)

An initial value problem specifies the value of the solution at a starting point, allowing the differential equation to be solved uniquely. Here, y(0) = 1 sets the initial condition, which is essential for applying Euler's method to approximate y(t) over the interval.

Step Size and Interval in Numerical Methods

The step size (Δt) determines the increments at which the solution is approximated, affecting accuracy and computational effort. The interval [0, T] defines the domain over which the solution is computed. Smaller step sizes generally yield more accurate results but require more calculations.