The Euler method is a numerical approximation technique used to solve ordinary differential equations. It is based on the concept of taking small steps along a curve and using the tangent line at each step to approximate the curve.
The Euler method works by dividing the interval of interest into smaller steps and using the derivative of the function at each step to approximate the value of the function at the next step. This process is repeated until the desired accuracy is achieved.
The Euler method is relatively simple to implement and does not require a lot of computational resources. It is also a good starting point for understanding more complex numerical methods for solving differential equations.
The Euler method can produce inaccurate results if the step size is too large or if the function has a high curvature. It also cannot handle stiff differential equations, where the behavior of the solution changes rapidly.
The accuracy of the Euler method can be improved by decreasing the step size and using a more sophisticated method for estimating the error, such as the Runge-Kutta method. Adding more terms to the Taylor series expansion used in the Euler method can also increase accuracy.