33–36. {Use of Tech} Computing Euler approximations Use a calculator or computer program to carry out the following steps.
c. Repeating parts (a) and (b) using half the time step used in those calculations, again find an approximation 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. Note the given time step \(\Delta t = 0.2\) and the target time \(T = 3\). The problem asks to repeat the Euler approximation using half the time step, so calculate the new time step as \(\Delta t = 0.1\).
3. Determine the number of steps needed to reach \(T=3\) using the new time step: \(n = \frac{T}{\Delta t} = \frac{3}{0.1} = 30\) steps.
4. Set up the Euler iteration formula: \(y_{k+1} = y_k + \Delta t \cdot f(t_k, y_k)\), where \(f(t, y) = 6 - 2y\). Start with \(y_0 = -1\) at \(t_0 = 0\).
5. Perform the iteration 30 times, updating \(y\) at each step using the formula, to approximate \(y(3)\) with the smaller time step.

Key Concepts

Euler's Method for Numerical Approximation

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. The formula is y_{n+1} = y_n + Δt * f(t_n, y_n), where Δt is the time step.

Effect of Step Size on Accuracy

The step size Δt in Euler's method controls the increments at which the solution is approximated. Smaller step sizes generally yield more accurate results because they better capture the curve's behavior, but require more computations. Halving Δt typically improves the approximation by reducing local truncation error.

Analytical Solution of Differential Equations

An analytical solution is an exact formula that satisfies the differential equation and initial conditions. For y' = 6 - 2y with y(0) = -1, the solution y(t) = 3 - 4e^{-2t} provides a benchmark to compare numerical approximations and assess their accuracy.