Euler Method for Solving First Order Differential Equation

Given $y' = y$ with an initial condition $y(0) = 1$ and step size $h = 0.01$, use the Euler method to approximate the solution of the differential equation.

The Euler method is a fundamental numerical technique used to approximate solutions of ordinary differential equations (ODEs). It provides a simple way to step through a solution, making it ideal for problems where an exact solution is difficult or impossible to find analytically. When using the Euler method, we begin at a known value (the initial condition) and keep taking steps forward in time (or the independent variable) to calculate subsequent values. The step size, denoted $h$, determines how far each step takes, with smaller values generally increasing accuracy but also computation time.

In this problem, the initial value problem is straightforward because the differential equation $y' = y$ has a well-known and simple exponential solution. However, using Euler's method helps illustrate the concept of numerical approximation, especially with a small step size like 0.01, which ensures a finer approximation to the true curve.

While the Euler method is not the most accurate compared to other advanced methods like the Runge-Kutta methods, it serves as a stepping stone for understanding more complex numerical methods.