Trapezoidal Method Modified Euler

This method applies a centered difference approximation for the derivative and an average value for / on the right-hand side. The finite difference analog is centered about the point. That is, the differential equation is discretized at and f y + i2, t +in) is approximated by the average of the end-point values. Therefore, this method is an average of the forward and backward Euler methods, and the discrete approximation appears in the following equation  

This equation is implicit in If/is nonlinear, then a nonlinear equation must 

Looking again at the exponential test problem given by Equation 5.8 and applying the trapezoidal method, the following recurrence results  

Applying the trapezoidal method to the ODE-IVP of Equation 5.6 leads to the following  

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