# Calculus 2 : Euler's Method

## Example Questions

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### Example Question #1 : New Concepts

Suppose we have the following differential equation with the initial condition:

Use Euler's method to approximate , using a step size of .

Explanation:

We start at x = 0 and move to x=2, with a step size of 1. Essentially, we approximate the next step by using the formula:

.

So applying Euler's method, we evaluate using derivative:

And two step sizes, at x = 1 and x = 2.

And thus the evaluation of p at x = 2, using Euler's method, gives us p(2) = 2.

### Example Question #2 : New Concepts

Approximate  by using Euler's method on the differential equation

with initial condition  (which has the solution ) and time step

Explanation:

Using Euler's method with  means that we use two iterations to get the approximation. The general iterative formula is

where each  is

is an approximation of , and , for this differential equation. So we have

So our approximation of  is

### Example Question #1 : Euler's Method

Use Euler's method to find the solution to the differential equation   at  with the initial condition  and step size .

Explanation:

Euler's method uses iterative equations to find a numerical solution to a differential equation.  The following equations

are solved starting at the initial condition and ending at the desired value.   is the solution to the differential equation.

In this problem,

Starting at the initial point

We continue using Euler's method until .  The results of Euler's method are in the table below.

Note: Solving this differential equation analytically gives a different answer, .  Future problems will explain this discrepancy.

### Example Question #2 : Euler's Method

Use Euler's method to find the solution to the differential equation   at  with the initial condition  and step size .

Explanation:

Euler's method uses iterative equations to find a numerical solution to a differential equation.  The following equations

are solved starting at the initial condition and ending at the desired value.   is the solution to the differential equation.

In this problem,

Starting at the initial point

We continue using Euler's method until .  The results of Euler's method are in the table below.

Note: Solving this differential equation analytically gives a different answer, . As the step size gets larger, Euler's method gives a more accurate answer.

### Example Question #3 : Euler's Method

Use Euler's method to find the solution to the differential equation   at  with the initial condition  and step size .

Explanation:

Euler's method uses iterative equations to find a numerical solution to a differential equation.  The following equations

are solved starting at the initial condition and ending at the desired value.   is the solution to the differential equation.

In this problem,

Starting at the initial point

We continue using Euler's method until .  The results of Euler's method are in the table below.

### Example Question #4 : Euler's Method

Use Euler's method to find the solution to the differential equation   at  with the initial condition  and step size .

Explanation:

Euler's method uses iterative equations to find a numerical solution to a differential equation.  The following equations

are solved starting at the initial condition and ending at the desired value.   is the solution to the differential equation.

In this problem,

Starting at the initial point

We continue using Euler's method until .  The results of Euler's method are in the table below.

Note: Due to the simplicity of the differential equation, Euler's method finds the exact solution, even with a large step size,  Using a smaller step size is unnecessary and more time consuming.

### Example Question #5 : Euler's Method

Use Euler's method to find the solution to the differential equation   at  with the initial condition  and step size .

Explanation:

Euler's method uses iterative equations to find a numerical solution to a differential equation.  The following equations

are solved starting at the initial condition and ending at the desired value.   is the solution to the differential equation.

In this problem,

Starting at the initial point

We continue using Euler's method until .  The results of Euler's method are in the table below.

### Example Question #6 : Euler's Method

Use Euler's method to find the solution to the differential equation   at  with the initial condition  and step size .

Explanation:

Euler's method uses iterative equations to find a numerical solution to a differential equation.  The following equations

are solved starting at the initial condition and ending at the desired value.   is the solution to the differential equation.

In this problem,

Starting at the initial point

We continue using Euler's method until .  The results of Euler's method are in the table below.

### Example Question #7 : Euler's Method

Use Euler's method to find the solution to the differential equation   at  with the initial condition  and step size .

Explanation:

Euler's method uses iterative equations to find a numerical solution to a differential equation.  The following equations

are solved starting at the initial condition and ending at the desired value.   is the solution to the differential equation.

In this problem,

Starting at the initial point

We continue using Euler's method until .  The results of Euler's method are in the table below.

### Example Question #8 : Euler's Method

Use Euler's method to find the solution to the differential equation   at  with the initial condition  and step size .

Explanation:

Euler's method uses iterative equations to find a numerical solution to a differential equation.  The following equations

are solved starting at the initial condition and ending at the desired value.   is the solution to the differential equation.

In this problem,

Starting at the initial point

We continue using Euler's method until .  The results of Euler's method are in the table below.

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