Domanda del libro di testo
What integral equation is equivalent to the initial value problem y' = f(x), y(x₀) = y₀?
6
views
Ch. 9 - First-Order Differential Equations
Hass15th EditionThomas' CalculusISBN: 9780137616077
Non è quello che usi tu?Cambia libro di testoScegli un libro di testo diverso
TOC.tableOfContents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Capitolo 9, Problema 9.1.18
Using Euler’s Method
In Exercises 15–20, use Euler’s method to calculate the first three approximations to the given initial value problem for the specified increment size. Calculate the exact solution and investigate the accuracy of your approximations. Round your results to four decimal places.
y' = y²(1+2x), (y-1) = 1, dx = 0.5
Guida verificata passo dopo passo1
Identify the initial value problem (IVP) and the given parameters: the differential equation is \(y' = y^{2}(1 + 2x)\), the initial condition is \(y(1) = 1\), and the step size is \(\Delta x = 0.5\).
Set up Euler's method formula: given a current point \((x_n, y_n)\), the next approximation is \(y_{n+1} = y_n + \Delta x \cdot f(x_n, y_n)\), where \(f(x, y) = y^{2}(1 + 2x)\) is the derivative function.
Calculate the first approximation \(y_1\) at \(x_1 = x_0 + \Delta x = 1 + 0.5 = 1.5\) using the initial values: \(y_1 = y_0 + 0.5 \times y_0^{2}(1 + 2x_0)\).
Calculate the second approximation \(y_2\) at \(x_2 = 2.0\) using the previous approximation: \(y_2 = y_1 + 0.5 \times y_1^{2}(1 + 2x_1)\).
Calculate the third approximation \(y_3\) at \(x_3 = 2.5\) similarly: \(y_3 = y_2 + 0.5 \times y_2^{2}(1 + 2x_2)\). After these, find the exact solution of the differential equation with the initial condition and compare the values to analyze the accuracy.
Risposta video verificata per un problema simile:
Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Durata del video:
8mConcetti chiave
Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.
Euler's Method
Euler's Method is a numerical technique to approximate solutions of first-order differential equations. Starting from an initial value, it uses the slope given by the differential equation to estimate the function's value at successive points, incrementing by a fixed step size. This method is simple but can accumulate errors depending on the step size.
Video consigliato:
07:33
Euler's Method
Initial Value Problem (IVP)
An Initial Value Problem specifies a differential equation along with a starting point (initial condition) for the solution. The initial condition, such as y(1) = 1, provides a specific value that the solution must satisfy, allowing for a unique solution curve to be approximated or found exactly.
Video consigliato:
Percorso guidato
05:03Initial Value Problems
Exact Solution and Error Analysis
The exact solution is the precise function satisfying the differential equation and initial condition. Comparing Euler's approximations to the exact solution helps assess the accuracy of the numerical method. Error analysis involves calculating the difference between approximate and exact values, often influenced by step size and method order.
Video consigliato:
Percorso guidato
04:57Determining Error and Relative Error
Pratica correlata
Domanda del libro di testo
Using Euler’s Method
In Exercises 15–20, use Euler’s method to calculate the first three approximations to the given initial value problem for the specified increment size. Calculate the exact solution and investigate the accuracy of your approximations. Round your results to four decimal places.
y' = 2y/x, y(1) = -1, dx = 0.5
25
views
Domanda del libro di testo
Solving Initial Value Problems
Solve the initial value problems in Exercises 15–20.
dy/dx + xy = x, y(0) = -6
5
views
Domanda del libro di testo
First-Order Linear Equations
Solve the differential equations in Exercises 1–14.
e²ˣy' + 2e²ˣ y = 2x
31
views
Domanda del libro di testo
Is either of the following equations correct? Give reasons for your answers.
a. (1/cosx) ∫ cos x dx = tan x + C
b. (1/cosx) ∫ cos x dx = tan x + C / cos x
25
views
Domanda del libro di testo
In Exercises 39–42, use Euler’s method with the specified step size to estimate the value of the solution at the given point x*. Find the value of the exact solution at x*.
y' = 1 + y², y(0) = 0, dx = 0.1, x* = 1
19
views