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Ch. 9 - First-Order Differential Equations
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Hass 15th EditionCh. 9 - First-Order Differential Equations문제 9.1.16
9장, 문제 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

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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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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. 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.
추천 영상:
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.
추천 영상:
05:03
Initial 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.
추천 영상:
04:57
Determining Error and Relative Error
관련 실천
교과서 질문

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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교과서 질문

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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교과서 질문

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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교과서 질문

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.

15
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교과서 질문

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)?

40
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교과서 질문

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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