7.5 Approximating Solutions Using Euler’s Method
3 min read•february 15, 2024
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What is Euler's Method?
Euler’s method is a way to find the of functions based on a given differential equation and an initial condition. We can approximate a function as a set of line segments using Euler’s method. 📈
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Before introducing this idea, it is necessary to understand two basic ideas.
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This information allows us to do an algorithmic process to approximate function values when given a differential equation and an initial condition.
To showcase this method, let’s consider the following differential equation with a consequent initial condition:
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Let’s say we want to approximate y(7). We will create a table that essentially creates that line-segment link in Fig. 7.1:
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Notice that we can fill in the rest of the table and continue the process to get closer and closer to x = 7. Also notice that the change in x is a constant value (which is typically called the ).
We use the differential equation to find the at the given point and use Eq. 41 to find the change in y:
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We can then find the new value of y by adding the change in y from the original y value:
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We can then fill in the rest of the table:
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Note that as the step size approaches zero, the approximation becomes more and more exact. As an exercise, find an approximate value for y(9). This means that x = 7 corresponds to y = 249, which is our approximate solution. 😃
Shorthand Summary of the Method
is a for approximating a solution to a differential equation. It is a simple and easy-to-implement method that is widely used in physics, engineering, and other fields.
is based on the idea of approximating the of a differential equation by a sequence of straight lines. The method starts with an initial point on the , and then generates a sequence of points by moving along the at each point. The is determined by the of the at that point, which is given by the of the solution function. 🧗
The basic procedure for is as follows:
- Start with an initial point (x0, y0) on the .
- Use the of the solution function (dy/dx = f(x, y)) to find the at the initial point.
- Use the to estimate the next point on the by moving a small step in the x-direction, called h, and adding the times h to the y-value at the initial point.
- Repeat steps 2 and 3 to generate a sequence of points on the .
The main advantage of is that it is simple to implement and understand. It is a good method for approximating a solution when an exact solution is not available. However, it is not very accurate and it may produce large errors when the step size is large.
Practice
Using Euler’s method, approximate the value of y(2) using a step size of 0.25 given the following:
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Then find the in the approximation by directly solving for y(2) by using a calculator. Approximate y(2) again using a step size of 0.2 and compare the in this approximation to the original . 🐪
Answer
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Key Terms to Review (10)
Absolute Error
: Absolute error measures the difference between an estimated value and its corresponding true value without considering directionality. It gives the magnitude of the discrepancy between the two values.Derivative
: A derivative represents the rate at which a function is changing at any given point. It measures how sensitive one quantity is to small changes in another quantity.Euler's Method
: Euler's Method is a numerical approximation technique used to estimate the value of a function at certain points when its derivative is known. It involves using small steps and linear approximations based on the slope at each point.First-Order Numerical Procedure
: A first-order numerical procedure is an algorithm used to approximate solutions to first-order ordinary differential equations. It involves dividing the interval into small steps and using iterative calculations to estimate values at each step.Function Approximation
: Function approximation is the process of estimating the value of an unknown function using known data points or simpler functions. It allows us to make predictions or simplify complex calculations.Numerical Values
: Numerical values are specific numbers used in calculations or measurements. They can be exact or approximate, and they provide quantitative information.Slope
: The slope refers to the steepness of a line. It measures how much a line rises or falls for every unit it moves horizontally.Solution Curve
: A solution curve is a graph that represents the behavior of a differential equation. It shows how the dependent variable changes over time or another independent variable.Step-size
: Step-size refers to the size or interval between consecutive values when working with sequences, series, or iterative processes. It determines how finely we divide our domain into smaller parts for analysis.Tangent Line
: A tangent line is a straight line that touches a curve at only one point without crossing through it. In calculus, we use tangent lines to approximate curves and find instantaneous rates of change.7.5 Approximating Solutions Using Euler’s Method
3 min read•february 15, 2024
What is Euler's Method?
Euler’s method is a way to find the of functions based on a given differential equation and an initial condition. We can approximate a function as a set of line segments using Euler’s method. 📈
Before introducing this idea, it is necessary to understand two basic ideas.
This information allows us to do an algorithmic process to approximate function values when given a differential equation and an initial condition.
To showcase this method, let’s consider the following differential equation with a consequent initial condition:
Let’s say we want to approximate y(7). We will create a table that essentially creates that line-segment link in Fig. 7.1:
Notice that we can fill in the rest of the table and continue the process to get closer and closer to x = 7. Also notice that the change in x is a constant value (which is typically called the ).
We use the differential equation to find the at the given point and use Eq. 41 to find the change in y:
We can then find the new value of y by adding the change in y from the original y value:
We can then fill in the rest of the table:
Note that as the step size approaches zero, the approximation becomes more and more exact. As an exercise, find an approximate value for y(9). This means that x = 7 corresponds to y = 249, which is our approximate solution. 😃
Shorthand Summary of the Method
is a for approximating a solution to a differential equation. It is a simple and easy-to-implement method that is widely used in physics, engineering, and other fields.
is based on the idea of approximating the of a differential equation by a sequence of straight lines. The method starts with an initial point on the , and then generates a sequence of points by moving along the at each point. The is determined by the of the at that point, which is given by the of the solution function. 🧗
The basic procedure for is as follows:
- Start with an initial point (x0, y0) on the .
- Use the of the solution function (dy/dx = f(x, y)) to find the at the initial point.
- Use the to estimate the next point on the by moving a small step in the x-direction, called h, and adding the times h to the y-value at the initial point.
- Repeat steps 2 and 3 to generate a sequence of points on the .
The main advantage of is that it is simple to implement and understand. It is a good method for approximating a solution when an exact solution is not available. However, it is not very accurate and it may produce large errors when the step size is large.
Practice
Using Euler’s method, approximate the value of y(2) using a step size of 0.25 given the following:
Then find the in the approximation by directly solving for y(2) by using a calculator. Approximate y(2) again using a step size of 0.2 and compare the in this approximation to the original . 🐪
Answer
Key Terms to Review (10)
Absolute Error
: Absolute error measures the difference between an estimated value and its corresponding true value without considering directionality. It gives the magnitude of the discrepancy between the two values.Derivative
: A derivative represents the rate at which a function is changing at any given point. It measures how sensitive one quantity is to small changes in another quantity.Euler's Method
: Euler's Method is a numerical approximation technique used to estimate the value of a function at certain points when its derivative is known. It involves using small steps and linear approximations based on the slope at each point.First-Order Numerical Procedure
: A first-order numerical procedure is an algorithm used to approximate solutions to first-order ordinary differential equations. It involves dividing the interval into small steps and using iterative calculations to estimate values at each step.Function Approximation
: Function approximation is the process of estimating the value of an unknown function using known data points or simpler functions. It allows us to make predictions or simplify complex calculations.Numerical Values
: Numerical values are specific numbers used in calculations or measurements. They can be exact or approximate, and they provide quantitative information.Slope
: The slope refers to the steepness of a line. It measures how much a line rises or falls for every unit it moves horizontally.Solution Curve
: A solution curve is a graph that represents the behavior of a differential equation. It shows how the dependent variable changes over time or another independent variable.Step-size
: Step-size refers to the size or interval between consecutive values when working with sequences, series, or iterative processes. It determines how finely we divide our domain into smaller parts for analysis.Tangent Line
: A tangent line is a straight line that touches a curve at only one point without crossing through it. In calculus, we use tangent lines to approximate curves and find instantaneous rates of change.
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