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) = t/y, y(0) = 4; Δt = 0.1, T = 2; y(t) = √(t² + 16)

검증된 단계별 안내

Identify the given differential equation and initial condition: \(y'(t) = \frac{t}{y}\) with \(y(0) = 4\).

Note the exact solution provided: \(y(t) = \sqrt{t^2 + 16}\), which will be used to find the error at \(T = 2\).

Use Euler's method with step size \(\Delta t = 0.1\) to approximate \(y(2)\). Recall Euler's formula: \(y_{n+1} = y_n + \Delta t \cdot f(t_n, y_n)\), where \(f(t, y) = \frac{t}{y}\).

Starting from \(t_0 = 0\) and \(y_0 = 4\), iteratively compute \(y_1, y_2, \ldots, y_{20}\) (since \(\frac{2}{0.1} = 20\) steps) using Euler's method.

Calculate the error at \(t = 2\) by subtracting the Euler approximation \(y_{20}\) from the exact solution \(y(2) = \sqrt{2^2 + 16}\): \(\text{Error} = |y(2) - y_{20}|\).

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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 the function at discrete points, starting from an initial condition. The method updates the solution by adding the product of the derivative and the step size to the previous value.

Exact Solution of Differential Equations

The exact solution is an explicit formula that satisfies the differential equation and initial conditions. In this problem, y(t) = √(t² + 16) is the exact solution, providing the true value of y at any time t. Comparing the exact solution to numerical approximations helps assess 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 error quantifies the accuracy of the approximation and helps in understanding how step size and method choice affect results.