Differential Equations : Euler Method

Study concepts, example questions & explanations for Differential Equations

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Example Questions

Example Question #1 : Euler Method

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

Possible Answers:

Correct answer:

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 : Euler Method

Approximate  for  with time steps  and .

Possible Answers:

Correct answer:

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 : Euler Method

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

Possible Answers:

Correct answer:

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 #1 : Euler Method

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

Possible Answers:

Correct answer:

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 #2 : Euler Method

Use two steps of Euler's Method with  on

To three decimal places

Possible Answers:

4.420

4.413

4.428

4.425

4.408

Correct answer:

4.425

Explanation:

Euler's Method gives us

Taking one step

Taking another step

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