Approximating Solution Using Eulers Method

Given the differential equation $\frac{dY}{dX} = Y$ and the initial condition $Y(0) = 1$, use Euler's method with a step size of 0.5 to approximate $Y(1)$.

Euler's method is a simple yet powerful tool for numerically solving ordinary differential equations (ODEs), especially when an analytic solution is difficult or impossible to obtain. It approximates the solution by using a sequence of tangent line segments, generated by the derivative information provided by the differential equation. In this problem, the differential equation given is a first-order linear ODE. The method involves iterating using a fixed step size, starting from the initial condition provided. Each step gives an approximate value of the solution at a subsequent point by following the direction field dictated by the differential equation. This iterative process visually represents how the solution curves' slope guides the path of the solution, a concept closely tied to the geometric interpretation of differential equations.