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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.
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 : Numerical Solutions Of Ordinary Differential Equations
Approximate for with time steps and .
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 #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.
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
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 #1 : Euler Method
Use two steps of Euler's Method with on
To three decimal places
4.425
4.413
4.428
4.408
4.420
4.425
Euler's Method gives us
Taking one step
Taking another step
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