- 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

**Euler's Method** is a bunch of tangent line approximations stuck together. The basic idea is that you start with a differential equation and a point. You do a tangent line approximation to get a new point.

Then you use the new point to do another tangent line approximation.

You do this over and over until you get to the end (which will be specified in the problem). The catch is that, after the first tangent line, instead of drawing real tangent lines you'll be drawing pretend tangent lines. This will make more sense after a couple of examples.

Example 1

Let that passes through the point (0,0). Use Euler's Method to estimate |

Example 2

Let that passes through the point (0,0). Use Euler's Method to estimate |

Example 3

The function with |

Exercise 1

Let *y* be the solution to the differential equation

that passes through the point (0,0). Use Euler's Method to estimate *y*(1) with step size Δ *x* = 0.25.

Exercise 2

The function *y* = *f* (*x*) is a solution to the differential equation

and *f* (0) = 5. Estimate *y*(1) using Euler's method with four steps.

Exercise 3

To get from 0 to 1 in four steps means

Here's the final table:

We conclude that

*f* (1) ≅ 4.9247

Exercise 4

Let *y* = *f* (*x*) be the solution to the initial value problem

Use Euler's method to estimate *f* (4) using 5 steps.