Euler and Improved Euler Method for Approximating Solutions

Using the Euler and improved Euler techniques, approximate the values of $y$ given the initial conditions and a step size.

Numerical methods are pivotal when it comes to approximating solutions for differential equations, especially when analytical solutions are tough or impossible to find. The Euler method is one of the simplest numerical methods, which uses a straightforward algorithm to estimate the value of a function based on its previous value and the slope (given by the differential equation). In essence, it draws a tangent line at an initial point and uses this tangent to step forward a fixed amount, called the step size. However, because this method relies heavily on linear approximations, its accuracy decreases significantly with larger step sizes or more complex functions.

The improved Euler method, also known as Heun’s method or the modified Euler method, enhances the accuracy of the basic Euler method by incorporating an additional step: it calculates an initial approximation using the Euler method and then refines this estimate by averaging the slope at the beginning and end of the step. This dual-step approach reduces the error introduced by the linear approximation, leading to more accurate results compared to the simple Euler method, especially for smaller step sizes. When solving problems involving these methods, focus not just on applying the formulas, but also on understanding how step size affects accuracy and how these methods provide insights into the behavior of dynamic systems modeled by differential equations. Furthermore, this understanding builds a solid foundation for exploring more advanced numerical methods.