Domanda del libro di testo
Integral Equations
In Exercises 7–12, write an equivalent first-order differential equation
and initial condition for y.
y = ln x + ∫ₓᵉ √ (t² + (y(t))²) dt
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Ch. 9 - First-Order Differential Equations
Hass15th EditionThomas' CalculusISBN: 9780137616077
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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 9.1.16
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' = x(1-y), y(1) = 0, dx = 0.2
Guida verificata passo dopo passo1
Identify the differential equation and initial condition: \(y' = x(1 - y)\) with \(y(1) = 0\). Here, \(y'\) represents the derivative \(\frac{dy}{dx}\), and the initial point is at \(x = 1\) with \(y = 0\).
Set the step size \(\Delta x = 0.2\). Euler's method uses the formula \(y_{n+1} = y_n + \Delta x \cdot f(x_n, y_n)\), where \(f(x, y) = y'\) is the right-hand side of the differential equation.
Calculate the first approximation: Evaluate \(f(x_0, y_0) = x_0 (1 - y_0)\) at the initial point \((x_0, y_0) = (1, 0)\), then compute \(y_1 = y_0 + 0.2 \times f(x_0, y_0)\).
Calculate the second approximation: Update \(x_1 = x_0 + 0.2\), then evaluate \(f(x_1, y_1)\) using the previously found \(y_1\), and compute \(y_2 = y_1 + 0.2 \times f(x_1, y_1)\).
Calculate the third approximation: Update \(x_2 = x_1 + 0.2\), evaluate \(f(x_2, y_2)\), and compute \(y_3 = y_2 + 0.2 \times f(x_2, y_2)\). After these steps, find the exact solution by solving the differential equation analytically and compare the approximations to assess accuracy.
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Euler's Method
Euler's Method is a numerical technique to approximate solutions of first-order differential equations. Starting from an initial point, it uses the slope given by the differential equation to estimate the next value by moving a small step size along the tangent. This iterative process provides approximate values of the solution at discrete points.
Video consigliato:
07:33
Euler's Method
Initial Value Problem (IVP)
An Initial Value Problem specifies a differential equation along with a starting condition, or initial value, for the unknown function. The solution must satisfy both the differential equation and the initial condition, which anchors the solution curve at a specific point, enabling numerical or analytical methods to find the function.
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, often found analytically. Comparing Euler’s approximations to the exact solution helps assess accuracy. Error analysis involves measuring the difference between approximate and exact values, which typically decreases with smaller step sizes.
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Percorso guidato
04:57Determining Error and Relative Error
Pratica correlata
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′ = √x/y, y > 0, y(0) = 1, dx = 0.1, x* = 1
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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' = 2xexp(x²) , y(0) = 2, dx = 0.1, x* = 1
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Domanda del libro di testo
Write the formula for a logistic function that has values between y = 0 and y = 1, crosses the line y = 1/2 at x = 0, and has slope 5 at this point.
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Domanda del libro di testo
Use Euler’s method with dx = 0.2 to estimate y(1) if y′ = y and y(0) = 1. What is the exact value of y(1)?
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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' = 2y²(x-1), y(2) = -1/2, dx = 0.1, x* = 3
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