First order differential equations: Euler's method

Finally, we consider ?nding the numerical solution for a ?rst order differential equation given an initial value. We consider the Euler's method, which is the most basic procedure for ?nding the approximate solution of an initial value problem stated as:

dy/dx = f(x, y),

given the initial value

y(x₀) = y₀

where x₀ and y₀ are given parameters (numbers). The derivation of the Euler can be geometrical (by looking at the tangent) or by considering Taylor expansions and taking only ?rst order terms or alternatively by considering the basic de?nition of derivative in terms of limit theorem.

Let us consider the geometric form of the derivation for the Euler's method.