33–36. {Use of Tech} Computing Euler approximations Use a calculator or computer program to carry out the following steps.
b. Using the exact solution (also given), find the error in the approximation to y(T) (only at the right endpoint of the time interval).

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

검증된 단계별 안내

Identify the differential equation and initial condition: \(y'(t) = -2y\), with \(y(0) = 1\).

Set up Euler's method formula for approximating the solution: \(y_{n+1} = y_n + \Delta t \cdot f(t_n, y_n)\), where \(f(t, y) = -2y\).

Calculate the number of steps needed to reach \(T = 2\) using the step size \(\Delta t = 0.2\): \(N = \frac{T - 0}{\Delta t} = \frac{2}{0.2} = 10\) steps.

Iteratively apply Euler's method starting from \(y_0 = 1\) to compute \(y_1, y_2, \ldots, y_{10}\) using the formula from step 2.

Find the exact solution at \(T=2\) using the given formula \(y(2) = e^{-2 \cdot 2} = e^{-4}\), then calculate the error as \(|y_{10} - y(2)|\).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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 discrete points by moving along the slope given by y′(t). This method is simple but can accumulate error depending on the step size and interval length.

Exact Solution of Differential Equations

The exact solution is the precise function that satisfies the differential equation and initial condition. For y′(t) = -2y with y(0) = 1, the exact solution is y(t) = e⁻²ᵗ. Knowing the exact solution allows comparison with numerical approximations to evaluate their accuracy.

Error Analysis in Numerical Methods

Error analysis involves calculating the difference between the numerical approximation and the exact solution at a specific point, here at t = T. This helps assess the accuracy of the approximation and understand how step size and method choice affect the result.