Approximating Differential Equations Using Euler Method

Given $y' = y - x^2$ with initial condition $y(0) = 0$ and step size $h = 0.1$, use the Euler method to approximate the solution of the differential equation.

The Euler method is a fundamental numerical technique used for approximating solutions to differential equations, particularly when analytical solutions are difficult or impossible to obtain. This method is especially useful in engineering and scientific applications where such equations frequently arise. The Euler method provides an iterative approach to estimating the value of a function's next step using the current point's derivative information, making it a straightforward yet powerful tool in numerical analysis.

In this particular problem, we are given a differential equation with an initial condition and tasked to approximate the solution using a specific step size. The Euler method is applied here, incrementing through the interval defined by the step size, and at each point, the derivative is used to project the function's next value. The goal is to gain an understanding of how the solution behaves over the given domain.

Conceptually, the Euler method's strength and weakness lie in its simplicity. It is easy to implement but may require smaller step sizes for higher accuracy, which increases computational cost. This method introduces errors at each step, known as local truncation errors, but these can accumulate, leading to a global truncation error over the entire computational interval. By exploring and applying the Euler method to this problem, students get to appreciate the balance between computational simplicity and the trade-off with accuracy, a critical aspect of numerical methods in differential equations.