Solving Differential Equations:One step Methods

A lot of problems in science and engineering involve solving differential equations pendulums, RLC circuits, diffusion and so on. An ordinary differential equation can be expressed as: 

Generally solution of this equation is of the form:

Here ‘phi‘ is the increment function or slope and h is the step size.  Thus, an estimate of slope is used to extrapolate the new value y(i+1) when the previous value y(i) is known, over a distance h.  The methods which are based on above equation are called one step methods. They all only differ by the definition of increment function.

Solved Example:
Calculate y(2) for in the interval t=[0,2]. Given y(0)=1, for
Consider h=1, here
Using Euler Method:

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