Numerical Computation (Euler method)

In summary, the Euler method is a numerical approximation technique for solving ordinary differential equations. It works by taking small steps along a curve and using the tangent line to approximate the curve. Its advantages include simplicity and low computational requirements, making it a good starting point for understanding more complex methods. However, it has limitations in terms of accuracy and cannot handle certain types of differential equations. To improve accuracy, one can decrease the step size, use a different error estimation method, or add more terms to the Taylor series expansion.
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Homework Statement


Differential equations are important in physics.Consider Solving the differential equation dy/dx = Σ(from i to N) (ai) (x^i)
Using Euler ,Midpoint ,4th order Runge -Kutta methods.For each of these methods what is the largest value of N that would lead to an exact solution ?

Homework Equations


The Attempt at a Solution


Homework Statement


Homework Equations


The Attempt at a Solution

 
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  • #2
Well, what are you having difficulty with?
 
  • #3
I simply do not have an idea what i have to do
I don't understand anything. What are the conditions for exact solution ??
I don't know anything.
 

1. What is the Euler method?

The Euler method is a numerical approximation technique used to solve ordinary differential equations. It is based on the concept of taking small steps along a curve and using the tangent line at each step to approximate the curve.

2. How does the Euler method work?

The Euler method works by dividing the interval of interest into smaller steps and using the derivative of the function at each step to approximate the value of the function at the next step. This process is repeated until the desired accuracy is achieved.

3. What are the advantages of using the Euler method?

The Euler method is relatively simple to implement and does not require a lot of computational resources. It is also a good starting point for understanding more complex numerical methods for solving differential equations.

4. What are the limitations of the Euler method?

The Euler method can produce inaccurate results if the step size is too large or if the function has a high curvature. It also cannot handle stiff differential equations, where the behavior of the solution changes rapidly.

5. How can I improve the accuracy of the Euler method?

The accuracy of the Euler method can be improved by decreasing the step size and using a more sophisticated method for estimating the error, such as the Runge-Kutta method. Adding more terms to the Taylor series expansion used in the Euler method can also increase accuracy.

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