How to Solving Ordinary Differential Equation (ODE)

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Solving ODE using Excel

For this section of introductory tutorial, we will focus first to the first degree and first order ODE. This is because we can always put higher degree ODE into several equations of the first degree ODE.

Solving an ODE means, we have an ordinary differential equation How to Solving Ordinary Differential Equation (ODE) and we want to integrate to get the solution equation How to Solving Ordinary Differential Equation (ODE)

Given:

1. ODE How to Solving Ordinary Differential Equation (ODE)

2. Upper and lower boundary of domain How to Solving Ordinary Differential Equation (ODE) ,

3. Initial value (usually How to Solving Ordinary Differential Equation (ODE) , that is the value of How to Solving Ordinary Differential Equation (ODE) when How to Solving Ordinary Differential Equation (ODE) )

Find the value of How to Solving Ordinary Differential Equation (ODE) within domain How to Solving Ordinary Differential Equation (ODE)

Though theoretically we can solve ODE using integration, in many practical cases, it is very difficult to solve the integration because the existence of the dependent variable How to Solving Ordinary Differential Equation (ODE) in the right hand side of the ODE How to Solving Ordinary Differential Equation (ODE) . We need to go around by solving the integration through an approximation of numerical solution within a range of domain How to Solving Ordinary Differential Equation (ODE) . To solve the ODE numerically, the most widely used solution is using How to Solving Ordinary Differential Equation (ODE) order Runge-Kutta method. Higher order approximation is possible but may be unstable to small variation of initial condition and may lead to a much higher computational cost.

Runge-Kutta (RK) method is a one-step iterative procedure to approximate integration of ODE How to Solving Ordinary Differential Equation (ODE) . Runge-Kutta methods have several advantages compared to many other methods of integration (such as Monte Carlo , Adam-Moulton methods, etc.):

  1. No other function is used rather than How to Solving Ordinary Differential Equation (ODE) is used for the integration,
  2. No additional starting values is needed
  3. No additional differentiation is needed

The basis of Runge-Kutta methods is Taylor series expansion

How to Solving Ordinary Differential Equation (ODE)

We can truncate these expansion terms up to a certain order of error term because for small value of step How to Solving Ordinary Differential Equation (ODE) , higher power of How to Solving Ordinary Differential Equation (ODE) would be very small and negligible.

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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\