Skip to main content
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 9.1.39
9장, 문제 9.1.39

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

검증된 단계별 안내
1
Identify the differential equation and initial condition: \(y' = 2x \exp(x^2)\) with \(y(0) = 2\).
Set up Euler's method formula: \(y_{n+1} = y_n + f(x_n, y_n) \cdot \Delta x\), where \(f(x, y) = 2x \exp(x^2)\) and \(\Delta x = 0.1\).
Calculate the number of steps needed to reach \(x^* = 1\) starting from \(x_0 = 0\) using \(\text{steps} = \frac{x^* - x_0}{\Delta x} = \frac{1 - 0}{0.1} = 10\) steps.
Iteratively apply Euler's method: for each step \(n\) from 0 to 9, compute \(y_{n+1} = y_n + 2x_n \exp(x_n^2) \cdot 0.1\) and update \(x_{n+1} = x_n + 0.1\).
To find the exact solution at \(x = 1\), integrate the differential equation: \(y = \int 2x \exp(x^2) \, dx + C\). Use substitution \(u = x^2\) to solve the integral, then apply the initial condition \(y(0) = 2\) to find \(C\).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
9m
도움이 되었나요?

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 condition, it uses the slope given by the differential equation to estimate the next value by moving in small steps (dx). This iterative process continues until reaching the desired point.
추천 영상:
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 solution curve is uniquely determined by this initial value, allowing numerical methods like Euler's to approximate the solution over an interval.
추천 영상:
05:03
Initial Value Problems

Exact Solution of Differential Equations

The exact solution is an explicit formula that satisfies the differential equation and initial condition. Finding it often involves integration or known solution techniques. Comparing the exact solution to numerical approximations helps assess the accuracy of methods like Euler's.
추천 영상:
04:00
Solutions to Basic Differential Equations
관련 실천
교과서 질문

First-Order Linear Equations

Solve the differential equations in Exercises 1–14.


tan θ dr/dθ + r = sin²θ, 0 < θ < π/2

17
views
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

38
views
교과서 질문

Carbon monoxide pollution An executive conference room of a corporation contains 4500 ft³ of air initially free of carbon monoxide. Starting at time t = 0, cigarette smoke containing 4% carbon monoxide is blown into the room at the rate of 0.3 ft³/min. A ceiling fan keeps the air in the room well circulated and the air leaves the room at the same rate of 0.3 ft³/min. Find the time when the concentration of carbon monoxide in the room reaches 0.01%.

29
views
교과서 질문

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

19
views
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
views
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

24
views