Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 9: Problem 12
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. $$y^{\prime}=x(1-y), \quad y(1)=0, \quad d x=0.2$$
Short Answer
Expert verified
Euler's method gives approximations: 0.2, 0.392, 0.5855 at x = 1.2, 1.4, 1.6. Errors with exact values are generally small.
Step by step solution
01
Understand Euler's Method
Euler's method is a numerical technique for solving ordinary differential equations (ODEs) with a given initial value. It uses a given step size to approximate the solution iteratively. For an equation \( y' = f(x, y) \) with an initial condition \( y(x_0) = y_0 \), the iterative formula is:\[y_{n+1} = y_n + dx \cdot f(x_n, y_n)\]where \( dx \) is the increment size, \( x_n \) and \( y_n \) are the current point coordinates.
02
Set Initial Values
The initial condition provided is \( y(1) = 0 \), so we set \( x_0 = 1 \) and \( y_0 = 0 \). The step size \( dx \) is given as 0.2.
03
First Approximation
Using the Euler's method formula with the initial values:\[y_1 = y_0 + dx \cdot f(x_0, y_0) = 0 + 0.2 \cdot (1 \cdot (1 - 0)) = 0.2\]Thus, \( y_1 = 0.2 \) when \(x_1 = 1.2\).
04
Second Approximation
Proceed with the next iteration using \( x_1 = 1.2 \) and \( y_1 = 0.2 \):\[y_2 = y_1 + dx \cdot f(x_1, y_1) = 0.2 + 0.2 \cdot (1.2 \cdot (1 - 0.2)) = 0.392\]Thus, \( y_2 = 0.392 \) when \( x_2 = 1.4 \).
05
Third Approximation
Continue iterating with \( x_2 = 1.4 \) and \( y_2 = 0.392 \):\[y_3 = y_2 + dx \cdot f(x_2, y_2) = 0.392 + 0.2 \cdot (1.4 \cdot (1 - 0.392)) = 0.5855\]Thus, \( y_3 = 0.5855 \) when \( x_3 = 1.6 \).
06
Calculate the Exact Solution
The differential equation is \( y' = x(1-y) \). To find the exact solution, separate variables and integrate:\[\frac{dy}{1-y} = x \, dx\]Integrating both sides gives:\[-\ln|1-y| = \frac{x^2}{2} + C\]Solving for \( y \), we get:\[y = 1 - e^{- rac{x^2}{2} + C}\]Using the initial condition \( y(1) = 0 \):\[1 - e^{- rac{1}{2} + C} = 0 \rightarrow e^{C} = e^{ rac{1}{2}}\]Thus, the exact solution is:\[y = 1 - e^{ rac{1-x^2}{2}}\]
07
Evaluate Exact Solution at Approximated Points
Using the exact solution \( y = 1 - e^{(1-x^2)/2} \), calculate at \( x = 1.2, 1.4, 1.6 \):- \( x = 1.2 \), \( y_{exact} = 1 - e^{(1-1.2^2)/2} = 0.1987 \)- \( x = 1.4 \), \( y_{exact} = 1 - e^{(1-1.4^2)/2} = 0.3888 \)- \( x = 1.6 \), \( y_{exact} = 1 - e^{(1-1.6^2)/2} = 0.5783 \)
08
Investigate Accuracy of Approximations
Compare Euler's method approximations with exact values:- At \( x = 1.2 \), Euler's \( y_1 = 0.2 \), Exact \( y = 0.1987 \), Error = 0.0013.- At \( x = 1.4 \), Euler's \( y_2 = 0.392 \), Exact \( y = 0.3888 \), Error = 0.0032.- At \( x = 1.6 \), Euler's \( y_3 = 0.5855 \), Exact \( y = 0.5783 \), Error = 0.0072.
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Numerical Approximation
Numerical approximation is a method used to find an estimated solution to complex mathematical problems that cannot be easily solved by analytical methods. In many cases, exact solutions are either unknown or too difficult to obtain.
With numerical approximation, iterative methods like Euler's Method are applied to provide solutions that are close enough to be practically useful. This approach divides the problem into smaller, more manageable steps, each refined by calculations based on current knowledge or guesses.
With numerical approximation, iterative methods like Euler's Method are applied to provide solutions that are close enough to be practically useful. This approach divides the problem into smaller, more manageable steps, each refined by calculations based on current knowledge or guesses.
- Euler's Method is one such approach, involving calculating future values based on current data.
- It helps explore solutions for ordinary differential equations (ODEs) iteratively.
Ordinary Differential Equation
Ordinary Differential Equations (ODEs) are equations that relate a function to its derivatives. ODEs are crucial in modeling various real-world phenomena where change is continuous, such as population growth, heat transfer, and motion dynamics.
A standard form of an ODE is expressed as \( y' = f(x, y) \), where \( y' \) is the derivative of the function \( y \) concerning an independent variable \( x \). In our exercise, the ODE given is \( y' = x(1-y) \).
Solving these equations often requires initial conditions to ensure a unique solution pathway, which Euler's Method assists with by offering a recursive relationship between known and unknown values of \( y \).
A standard form of an ODE is expressed as \( y' = f(x, y) \), where \( y' \) is the derivative of the function \( y \) concerning an independent variable \( x \). In our exercise, the ODE given is \( y' = x(1-y) \).
Solving these equations often requires initial conditions to ensure a unique solution pathway, which Euler's Method assists with by offering a recursive relationship between known and unknown values of \( y \).
- ODEs can represent various natural and physical processes that change over time.
- They are vital in predicting forward from an initial condition.
Initial Value Problem
An initial value problem (IVP) is a type of problem in mathematics where an ODE is paired with a specified value at a starting point. It's like taking a snapshot at time zero and using it to unravel the past and predict the future of the system described by ODE.
In solving an IVP, boundary constraints given at the start, such as \( y(1) = 0 \) in our exercise, are crucial. These conditions ground the solutions, making them not just theoretically sound but practically applicable by providing a tangible starting point.
In solving an IVP, boundary constraints given at the start, such as \( y(1) = 0 \) in our exercise, are crucial. These conditions ground the solutions, making them not just theoretically sound but practically applicable by providing a tangible starting point.
- They define the position from which our iterative or numerical methods begin.
- IVPs ensure that the path from known values leads logically to unknown solutions.
Error Analysis
Error analysis is the process of assessing the accuracy of solutions obtained through numerical methods like Euler's Method. Understanding the deviation between the approximated and exact solutions is critical for judging how reliable and practical these solutions are.
In the context of Euler's Method, error analysis involves comparing each step's approximate solution to the exact solution. This is illustrated by examining the discrepancies at defined points, such as those calculated in steps 7 and 8 of our exercise.
The main points of focus in error analysis include:
In the context of Euler's Method, error analysis involves comparing each step's approximate solution to the exact solution. This is illustrated by examining the discrepancies at defined points, such as those calculated in steps 7 and 8 of our exercise.
The main points of focus in error analysis include:
- Magnitude of deviation between Euler's approximation and the exact solution.
- Trends in error size, which can offer insights into adjusting step sizes or other considerations.
- Understanding that smaller steps generally yield more precise approximations, though at a computational cost.
