Eulers Method for Initial Value Problem

Estimate the value $Y(4)$ for the initial value problem: $y' - y = 0$ with $Y(0) = 1$ using Euler's method and a step size of 1.

Euler's method is a straightforward and commonly used numerical technique for solving initial value problems, particularly for differential equations where obtaining an exact analytical solution might be cumbersome or impossible. This method involves approximating the solution by advancing step by step, using the derivative's value and a given step size. In this problem, we start with a known initial condition, Y at 0 equals 1, and iterate forward to approximate the value at the desired point.

To understand the conceptual framework behind Euler's method, it's essential to visualize the solution of the differential equation as a curve in space. The slope of this curve at any point is given by the equation, which in this case is a simple first-order linear differential equation. By using Euler's method, we utilize this slope to step forward incrementally from the initial condition, effectively tracing out the curve piece by piece.

For this specific problem, you will implement Euler's method with a step size of 1, which is a relatively large step size and demonstrates how larger step sizes can lead to discrepancies from the true solution. It's important to note that Euler's method introduces some error, and the accuracy of the approximation can be improved by using smaller step sizes. This problem provides an opportunity to witness firsthand the balance between computational efficiency and accuracy in numerical solutions.