Textbook Question
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
Ch. 9 - First-Order Differential Equations
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- 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 Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 9, Problem 9.1.41
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
Verified step by step guidance1
Identify the differential equation and initial condition: \(y' = \frac{\sqrt{x}}{y}\) with \(y(0) = 1\).
Set the step size \(\Delta x = 0.1\) and the target point \(x^* = 1\). Determine the number of steps needed: \(n = \frac{x^* - 0}{\Delta x} = 10\) steps.
Apply Euler's method iteratively using the formula: \(y_{k+1} = y_k + \Delta x \cdot f(x_k, y_k)\), where \(f(x, y) = \frac{\sqrt{x}}{y}\).
Start with \(x_0 = 0\) and \(y_0 = 1\). For each step \(k\) from 0 to 9, compute \(y_{k+1}\) using the formula and update \(x_{k+1} = x_k + \Delta x\).
To find the exact solution at \(x^* = 1\), solve the differential equation analytically by separating variables and applying the initial condition, then evaluate the solution at \(x = 1\).
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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. It uses a step size to incrementally estimate the function's value by moving along the slope given by the differential equation. This method is especially useful when an exact solution is difficult to find.
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07:33
Euler's Method
Initial Value Problems (IVP)
An initial value problem specifies the value of the solution at a starting point, allowing the differential equation to be solved uniquely. Here, y(0) = 1 sets the initial condition, which is essential for applying Euler's method and finding the particular solution.
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05:03Initial Value Problems
Exact Solution of Differential Equations
The exact solution is an explicit formula that satisfies the differential equation and initial condition. Finding it involves techniques like separation of variables or integration. Comparing the exact solution to the Euler approximation helps assess the accuracy of the numerical method.
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04:00Solutions to Basic Differential Equations
Related Practice
Textbook Question
First-Order Linear Equations
Solve the differential equations in Exercises 1–14.
tan θ dr/dθ + r = sin²θ, 0 < θ < π/2
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
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
Textbook Question
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.