# Differential Equations : Euler Method

## Example Questions

### Example Question #1 : Numerical Solutions Of Ordinary Differential Equations

Use Euler's Method to calculate the approximation of  where  is the solution of the initial-value problem that is as follows.

Explanation:

Using Euler's Method for the function

first make the substitution of

therefore

where  represents the step size.

Let

Substitute these values into the previous formulas and continue in this fashion until the approximation for  is found.

Therefore,

### Example Question #2 : Numerical Solutions Of Ordinary Differential Equations

Approximate  for  with time steps  and .

Explanation:

Approximate  for  with time steps  and .

The formula for Euler approximations .

Plugging in, we have

Here we can see that we've gotten trapped on a horizontal tangent (a failing of Euler's method when using larger time steps). As the function is not dependent on t, we will continue to move in a horizontal line for the rest of our Euler approximations. Thus .

### Example Question #3 : Numerical Solutions Of Ordinary Differential Equations

Use Euler's Method to calculate the approximation of  where  is the solution of the initial-value problem that is as follows.

Explanation:

Using Euler's Method for the function

first make the substitution of

therefore

where  represents the step size.

Let

Substitute these values into the previous formulas and continue in this fashion until the approximation for  is found.

Therefore,

### Example Question #4 : Numerical Solutions Of Ordinary Differential Equations

Use the implicit Euler method to approximate  for , given that , using a time step of

Explanation:

In the implicit method, the amount to increase is given by , or in this case . Note, you can't just plug in to this form of the equation, because it's implicit:  is on both sides. Thankfully, this is an easy enough form that you can solve explicitly. Otherwise, you would have to use an approximation method like newton's method to find . Solving explicitly, we have  and .

Thus,

Thus, we have a final answer of

### Example Question #5 : Numerical Solutions Of Ordinary Differential Equations

Use two steps of Euler's Method with  on

To three decimal places

4.420

4.428

4.413

4.425

4.408