Ch. 9 - First-Order Differential Equations

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 9 - First-Order Differential Equations

Problem 9.1.16

Chapter 9, Problem 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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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.

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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Related Practice

Textbook Question

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

Textbook Question

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%.

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Textbook Question

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

Textbook Question

Use Euler’s method with dx = 1/3 to estimate y(2) if y′ = x sin y and y(0) = 1. What is the exact value of y(2)?

Textbook Question

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

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Textbook Question

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

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