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Magazine Chapter 8: Problem 11
Consider the initial-value problem $$ y^{\prime}=\sin \pi t, \quad y(0)=0 $$ Use Euler's Method with five steps to approximate \(y(1)\)
Short Answer
Expert verified
Using Euler's Method, \( y(1) \approx 0.5878 \).
Step by step solution
01
Understand Euler's Method
Euler's method is a numerical technique for finding approximate solutions to ordinary differential equations of the form \( y' = f(t, y) \). It uses the formula \( y_{n+1} = y_n + h imes f(t_n, y_n) \), where \( h \) is the step size, \( t_n \) is the current time, and \( y_n \) is the corresponding value of \( y \).
02
Initialize Values
We are given the differential equation \( y' = \sin(\pi t) \) and the initial condition \( y(0) = 0 \). The goal is to approximate \( y(1) \). We divide the interval from 0 to 1 into 5 equal steps, so \( h = \frac{1 - 0}{5} = 0.2 \). Thus, our initial values are \( t_0 = 0 \), \( y_0 = 0 \), and \( h = 0.2 \).
03
Apply Euler's Method Step by Step
For each step, we will calculate the next value of \( y \) using the formula \( y_{n+1} = y_n + h imes \sin(\pi t_n) \).
04
Sub-step: First Calculation
Calculate \( y_1 \): \( y_1 = y_0 + h \times \sin(\pi t_0) = 0 + 0.2 \times \sin(0) = 0 \). Now update \( t_1 = t_0 + h = 0.2 \).
05
Sub-step: Second Calculation
Calculate \( y_2 \): \( y_2 = y_1 + h \times \sin(\pi t_1) = 0 + 0.2 \times \sin(0.2\pi) \approx 0 + 0.2008 \approx 0.118 \). Update \( t_2 = t_1 + h = 0.4 \).
06
Sub-step: Third Calculation
Calculate \( y_3 \): \( y_3 = y_2 + h \times \sin(\pi t_2) = 0.118 + 0.2 \times \sin(0.4\pi) \approx 0.118 + 0.5878 \approx 0.2356 \). Update \( t_3 = t_2 + h = 0.6 \).
07
Sub-step: Fourth Calculation
Calculate \( y_4 \): \( y_4 = y_3 + h \times \sin(\pi t_3) = 0.2356 + 0.2 \times \sin(0.6\pi) \approx 0.2356 + 0.9511 \approx 0.4258 \). Update \( t_4 = t_3 + h = 0.8 \).
08
Sub-step: Fifth Calculation
Calculate \( y_5 \): \( y_5 = y_4 + h \times \sin(\pi t_4) = 0.4258 + 0.2 \times \sin(0.8\pi) \approx 0.4258 + 0.5878 \approx 0.5878 \). Update \( t_5 = t_4 + h = 1 \).
09
Final Result
The approximate value for \( y(1) \) using Euler's Method with five steps is \( y_5 \approx 0.5878 \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Numerical Methods
Numerical methods are powerful tools in mathematics that help us find approximate solutions to complex problems. They are especially useful when an equation does not have a closed-form solution, meaning it can't be solved exactly using traditional algebra.
Numerical methods can tackle a wide range of problems, including those involving integrals, roots of functions, and differential equations.
Euler's Method is a classic example of such numerical techniques. It approximates solutions by iteratively moving forward in small steps. The effectiveness of a numerical method often depends on the step size: a smaller step size usually yields more accurate results, but requires more calculations.
Numerical methods can tackle a wide range of problems, including those involving integrals, roots of functions, and differential equations.
Euler's Method is a classic example of such numerical techniques. It approximates solutions by iteratively moving forward in small steps. The effectiveness of a numerical method often depends on the step size: a smaller step size usually yields more accurate results, but requires more calculations.
Ordinary Differential Equations
Ordinary differential equations (ODEs) are fundamental to modeling dynamic systems where change is involved. An ODE involves a function and its derivatives, and it describes how the function evolves over time.
Many real-world phenomena, such as population growth, heat distribution, and fluid dynamics, are modeled with ODEs.
For a simple first-order ODE, like the one given in the exercise, the general form is \( y' = f(t, y) \). Here, \( y' \) represents the derivative of the function \( y \) with respect to \( t \). Solving such an equation means finding a function \( y(t) \) that satisfies the equation for given initial conditions.
Many real-world phenomena, such as population growth, heat distribution, and fluid dynamics, are modeled with ODEs.
For a simple first-order ODE, like the one given in the exercise, the general form is \( y' = f(t, y) \). Here, \( y' \) represents the derivative of the function \( y \) with respect to \( t \). Solving such an equation means finding a function \( y(t) \) that satisfies the equation for given initial conditions.
Initial-Value Problem
An initial-value problem is a type of problem for ODEs where the starting point of the function is specified. In essence, it means knowing the value of the function at a particular point and using this as a basis to solve the ODE.
For instance, in the problem \( y' = \sin \pi t, \ y(0) = 0 \), the initial condition \( y(0) = 0 \) is crucial. This condition determines the specific trajectory of the solution that Euler's Method will approximate.
Initial-value problems are common in physics and engineering, where you might know the initial state of a system and need to predict its future behavior. They allow the use of numerical methods like Euler鈥檚 to progress from a known point forward, calculating an approximate path that adheres to both the differential equation and the initial condition.
For instance, in the problem \( y' = \sin \pi t, \ y(0) = 0 \), the initial condition \( y(0) = 0 \) is crucial. This condition determines the specific trajectory of the solution that Euler's Method will approximate.
Initial-value problems are common in physics and engineering, where you might know the initial state of a system and need to predict its future behavior. They allow the use of numerical methods like Euler鈥檚 to progress from a known point forward, calculating an approximate path that adheres to both the differential equation and the initial condition.
