# Differential Equations

### Topics

## Introduction to Differential Equations - At A Glance:

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* = *x*^{2}.

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.

#### Example 1

Let Does Euler's method produce an over- or under-estimate for the value of |

#### Example 2

Let For which starting points will Euler's method produce underestimates and for which starting points will Euler's method produce overestimates? |

#### Exercise 1

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?

#### Exercise 2

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?

#### Exercise 3

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)?

#### Exercise 4

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)?

#### Exercise 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)?

#### Exercise 6

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)?

#### Exercise 7

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)?