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Numerical analysis is the branch of mathematics that deals with algorithms for solving mathematical problems numerically, rather than symbolically. Euler's method is a fundamental technique in numerical analysis, particularly for solving differential equations.
It allows us to approximate solutions, especially when finding an exact solution is hard or impossible. Using numerical methods, you can approximate an answer to any desired degree of accuracy by choosing the right method and step size. This approach is widely used in fields like engineering, physics, finance, and computer science to model complex systems.

An initial-value problem in mathematics refers to a differential equation accompanied by a specific starting point, known as the initial condition. In this case, the solution we are seeking is the function that satisfies the differential equation and fits the initial condition given.
A typical representation might be:

• \(y'(x) = f(x, y)\)
• Where \(y(x_0) = y_0\)

Here, \((x_0, y_0)\) represents the initial condition. The initial-value problem is foundational in dynamics and other areas where the state of a system at one point in time is used to predict its state at another. Using Euler's method helps in approximating the solution by breaking down the interval into smaller steps and iterating on each step using the differential equation.

Differential equations express relationships involving rates of change and the functions that govern these changes. Euler's method addresses ordinary differential equations (ODEs), which involve derivatives with respect to only one variable. Solving these equations is critical in many real-world applications where change is continuous.
Consider the exercise:

• The differential equation is \(y' = \sqrt{x+y}\).
• The initial condition is \(y(0) = 1\).

For such equations, Euler's numerical method provides an iterative way to approximate the function's values at discrete points. Each step uses the slope of the tangent line at the last calculated point to predict the next point. By adjusting the step size, you can achieve a more precise approximation.

Step size, often represented as \(\Delta x\), is a critical element in numerical methods like Euler's method. It defines how much you change your independent variable (\(x\)) from one iteration to the next.
This exercise calculates step size using:

• With \(n=4\), \(\Delta x = \frac{1.5 - 0}{4} = 0.375\)
• With \(n=6\), \(\Delta x = \frac{1.5 - 0}{6} = 0.25\)

Larger step sizes cover the zone quickly but can miss finer details and lead to less accurate approximations. Conversely, smaller step sizes increase accuracy by assessing more points but require more calculations.
Finding the right balance in step size ensures an accurate representation of the function while maintaining computational efficiency.