Formulation of The Method
Suppose that we want to approximate the solution of the initial value problem
Choose a value for the size of every step and set . Now, one step of the Euler method from to is
The value of is an approximation of the solution to the ODE at time : . The Euler method is explicit, i.e. the solution is an explicit function of for .
While the Euler method integrates a first-order ODE, any ODE of order N can be represented as a first-order ODE: to treat the equation
- ,
we introduce auxiliary variables and obtain the equivalent equation
This is a first-order system in the variable and can be handled by Euler's method or, in fact, by any other scheme for first-order systems.
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