- Topics At a Glance
- Differential Equations and Their Solutions
- Solutions to Differential Equations
- Solving Differential Equations
- Initial Value Problems
- More About Solutions
- Word Problems
- Slope Fields
- Slope Fields and Solutions
- Equilibrium Solutions
- Euler's Method
- Slopes (Again)
- Tangent Line Approximations (Again)
- The Scoop on Euler
- Accuracy and Usefulness of Euler's Method
- In the Real World
- I Like Abstract Stuff; Why Should I Care?
- How to Solve a Math Problem
Introduction to :
Things to remember about Euler's Method:
- Euler's Method gives only approximate values unless the function happens to be a straight line, in which case Euler's Method gives exact values of the function.
- Euler's method does NOT produce a formula. It generates a numerical solution; that is, approximate values of the function at certain points.
The error in an approximation is the difference between the approximation and the real value of the function.
We can only figure out the error if we know the real value of the function.
Sample Problem
Our first examples of Euler's method used the function y that satisfied the IVP
By thinking backwards we can figure out that
y = x2.
Now we can compute the real value of y at 1:
y(1) = (1)2 = 1.
This lets us figure out how bad our approximations were in some earlier examples.
Using a step size of 0.5 we estimated
y(1) ≅ 0.5.
This has an error of 0.5, since it's 0.5 off from the real answer y(1) = 1.
Using a step size of 0.25 we estimated
y(1) ≅ 0.75.
The error here is 0.25.
Using a step size of 0.2 we estimated
y(1) ≅ 0.8.
This has an error of 0.2.
Practice:
Let y = f (x) be a solution to the IVP
Does Euler's method produce an over- or under-estimate for the value of f (2)? |
looks like this:
Let y = f (x) be a solution to the IVP
For which starting points will Euler's method produce underestimates and for which starting points will Euler's method produce overestimates? |
Let y = f (x) be a solution to the IVP
What is the error when Euler's method is used to estimate f (2) with a step size of 0.5? With a step size of 0.25?
If f is concave up around x = a, is the tangent line to f at a above or below the graph of f? What about if f is concave down?
Let y = f (x) be a solution to the IVP
Does Euler's method produce an over- or under-estimate for the value of f (2)?
Let y = f (x) be a solution to the IVP
Does Euler's method produce an over- or under-estimate for the value of f (3.5)?
Let y = f (x) be a solution to the IVP
Does Euler's method produce an over- or under-estimate for the value of f (1)?
Let y = f (x) be a solution to the IVP
Does Euler's method produce an over- or under-estimate for the value of f (2)?
that passes through the point (1,1) is the line y = x. Let y = f (x) be a solution to the IVP
Does Euler's method produce an over- or under-estimate for the value of f (5.9)?
