Euler's Approximation to Differential Equations

Euler’s Method of approximating solutions to differential equations

Stacking Linearizations

Using linearizations that follow onto each-other one can determine a graph that closely follows the solution’s graph.

Using the first linearization at a point $x\_0$ we determine the linearization from point $(x\_0,y\_0)$ using $x\_1=x\_0+dx$

$$y\_1=L(x\_1)=y\_0+f(x\_0,y\_0)dx$$

Setting $x\_2=x\_1+dx$ we use the linearization of the solution curve through $(x\_1,y\_1)$ to obtain

$$y\_2=y\_1+f(x\_1,y\_1)dx$$

and so on. We are building an approximation to a solution by following the direction of the slope field of the differential equation.

Definition using $n$

$$x\_n=x\_n-1+dx$$ $$y\_n=y\_n-1+f(x\_n-1,y\_n-1)dx$$