33–36. {Use of Tech} Computing Euler approximations Use a calculator or computer program to carry out the following steps.
d. Compare the errors in the approximations to y(T).


y′(t) = 6 - 2y, y(0) = -1; Δt = 0.2, T = 3; y(t) = 3 - 4e⁻²ᵗ

Verified step by step guidance

1. Identify the differential equation and initial condition: \(y'(t) = 6 - 2y\), with \(y(0) = -1\).
2. Set the step size \(\Delta t = 0.2\) and the target time \(T = 3\). Calculate the number of steps as \(n = \frac{T}{\Delta t} = \frac{3}{0.2} = 15\) steps.
3. Use Euler's method formula to approximate \(y\) at each step: \(y_{k+1} = y_k + \Delta t \cdot f(t_k, y_k)\), where \(f(t, y) = 6 - 2y\). Start with \(y_0 = -1\) and iterate up to \(k = 15\).
4. Calculate the exact solution at \(T=3\) using the given formula: \(y(3) = 3 - 4e^{-2 \cdot 3} = 3 - 4e^{-6}\).
5. Find the error by subtracting the Euler approximation at \(T=3\) from the exact solution: \(\text{Error} = |y(3) - y_{15}|\). This will allow you to compare the accuracy of the Euler method with the exact value.

Key Concepts

Euler's Method

Euler's Method is a numerical technique to approximate solutions of first-order differential equations. It uses a step size Δt to iteratively estimate the value of y at successive points by applying the formula y_{n+1} = y_n + Δt * f(t_n, y_n). This method is especially useful when an exact solution is difficult to obtain.

Error Analysis in Numerical Methods

Error analysis involves comparing the approximate numerical solution to the exact solution to measure accuracy. The error at a point is the difference between the exact value y(T) and the Euler approximation. Understanding how step size affects error helps improve approximation quality.

Analytical Solution of Differential Equations

The analytical solution provides the exact formula for y(t), here given as y(t) = 3 - 4e^{-2t}. It serves as a benchmark to evaluate numerical methods. Knowing the exact solution allows direct computation of y(T) for error comparison with Euler's approximation.