Euler's Method (Analytical Interpretations)

1. If we approximate the derivative appearing in the differential equation at the point by a forward difference, we obtain

   

   Solving for yields the formula for the Euler's method.

2. Integrating the identity



between the limits and , we obtain



In particular, if and we get



Approximating the integral by a crude rule for numerical integration (length of the interval times the value of integrand at left end point) and identifying with , we obtain the Euler's method.