Euler's Method is a numerical method used to approximate the solutions of ordinary differential equations. It is based on the concept of using small steps to approximate a continuous curve.
Euler's Method works by dividing a continuous curve into small segments and approximating the slope of the curve at each point. These approximations are then used to calculate the next point on the curve, and this process is repeated until the desired accuracy is achieved.
Euler's Method is relatively simple to understand and implement, making it a popular choice for approximating solutions of differential equations. It also allows for easy adjustment of the step size, which can improve the accuracy of the approximation.
Euler's Method is a first-order method, meaning that it can introduce significant errors when used to approximate solutions of higher-order differential equations. It also requires a small step size to achieve high accuracy, which can lead to a large number of calculations and longer computation times.
The accuracy of Euler's Method can be demonstrated by comparing the results obtained using the method with the exact solution of the differential equation. This can be done by plotting the two solutions on a graph and observing the difference between them. Additionally, the error can be calculated and compared to the expected error for a given step size.