Solving Ordinary Differential Equation (ODE) using Euler Method

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Euler Method (Runge Kutta-1)

Euler method to integrate ordinary differential equation is sometimes also called Runge-Kutta order 1 (RK1) or Euler-Cauchy method.

Suppose we have ODE Solving Ordinary Differential Equation (ODE) using Euler Method then

Formula : Solving Ordinary Differential Equation (ODE) using Euler Method

Error term : Solving Ordinary Differential Equation (ODE) using Euler Method , correct up to the first order term of the Taylor series expansion is

Solving Ordinary Differential Equation (ODE) using Euler Method

Here is how the Euler method formula is obtained. When we want to find approximation formula that relates Solving Ordinary Differential Equation (ODE) using Euler Method and Solving Ordinary Differential Equation (ODE) using Euler Method for a very small step Solving Ordinary Differential Equation (ODE) using Euler Method , we can use Taylor series expansion of Solving Ordinary Differential Equation (ODE) using Euler Method

Solving Ordinary Differential Equation (ODE) using Euler Method

The big O-notation is the lowest order error term that the Taylor expansion differs from the Euler Method. At each step, the error is relatively big Solving Ordinary Differential Equation (ODE) using Euler Method , thus we need very small step size Solving Ordinary Differential Equation (ODE) using Euler Method to gain reasonable accuracy. Taking only the first two term of the Taylor expansion, and replace the derivative Solving Ordinary Differential Equation (ODE) using Euler Method , we get the Euler method formula (I emphasize that y is a function of x)

Solving Ordinary Differential Equation (ODE) using Euler Method

Notice that the second term of the right hand side [ Solving Ordinary Differential Equation (ODE) using Euler Method ] represents the slope at the beginning of the interval Solving Ordinary Differential Equation (ODE) using Euler Method as illustrated in the figure below

Euler method

The computation is using spreadsheet that can be downloaded here

Example:

Solving Ordinary Differential Equation (ODE) using Euler Method , set Solving Ordinary Differential Equation (ODE) using Euler Method , with initial condition Solving Ordinary Differential Equation (ODE) using Euler Method . The few first results and the graph of solution are also given below.

Solving Ordinary Differential Equation (ODE) using Euler Method Solving Ordinary Differential Equation (ODE) using Euler Method

Note that after Solving Ordinary Differential Equation (ODE) using Euler Method , the solution is not correct. See Comparison .

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See also: Numerical Excel tutorial , Dynamical System tutorial , Kardi Teknomo's Tutorial

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Teknomo, Kardi (2015) Solving Ordinary Differential Equation (ODE). http:\\people.revoledu.com\kardi\tutorial\ODE\