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Ch. 9 - Differential Equations
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 9 - Differential EquationsProblema 9.2.33b
Capitolo 9, Problema 9.2.33b

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

Guida verificata passo dopo passo
1
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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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.
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Euler's Method

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.
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Solutions to Basic Differential Equations

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.
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Percorso guidato
04:57
Determining Error and Relative Error
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Domanda del libro di testo

23–26. Stirred tank reactions For each of the following stirred tank reactions, carry out the following analysis.

b. Solve the initial value problem.


A 500-L tank is initially filled with pure water. A copper sulfate solution with a concentration of 20 g/L flows into the tank at a rate of 4 L/min. The thoroughly mixed solution is drained from the tank at a rate of 4 L/min.

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27–30. Predator-prey models Consider the following pairs of differential equations that model a predator-prey system with populations x and y. In each case, carry out the following steps.

b. Find the lines along which x'(t) = 0. Find the lines along which y'(t) = 0.


x′(t) = 2x − 4xy, y′(t) = −y + 2xy

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Domanda del libro di testo

17–20. Increasing and decreasing solutions Consider the following differential equations. A detailed direction field is not needed.


b. In what regions are solutions increasing? Decreasing?


y'(t) = (y−1)(1+y)

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Properties of stirred tank solutions


b. Verify that M(0) = M₀

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Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

b. Euler’s method is used to compute exact values of the solution of an initial value problem. 

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Domanda del libro di testo

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.


b. The solution of a stirred tank initial value problem always approaches a constant as t→∞

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