A numerical method to approximate solutions to differential equations with a given initial condition.

An equation that relates a function with its derivatives.

A point (x, y) on the solution curve of a differential equation.

The constant change in the x-value used in each iteration of Euler's Method.

It uses the derivative at a single point to approximate the function's value at the next point.

An approximate solution to a differential equation, usually in the form of a table of values.

Using the tangent line at a point to estimate the function's value nearby.

The absolute value of the difference between the approximate and exact values.

Smaller step sizes generally lead to more accurate approximations.

To approximate the solution of a differential equation when an analytical solution is difficult or impossible to find.

Euler's Method: Provides an approximate numerical solution. Exact Solution: Provides the precise analytical solution.

Smaller step size: More accurate, more computation. Larger step size: Less accurate, less computation.

Euler's Method: Uses the slope at the beginning of the interval. Improved Euler's Method: Averages slopes at the beginning and end of the interval for better accuracy.

Euler's Method: First-order, less accurate. Runge-Kutta: Higher-order, more accurate.

Numerical: Approximate, discrete values. Analytical: Exact, continuous function.

Local: Error in a single step. Global: Accumulated error over multiple steps.

Explicit: Uses known values to calculate the next value. Implicit: Uses unknown values, requiring solving an equation.

Forward: Uses the slope at the beginning of the interval. Backward: Uses the slope at the end of the interval.

Euler's Method: Uses only the first derivative term. Taylor Series: Uses multiple derivative terms for higher accuracy.

Euler's Method: Approximates using the slope at the beginning of the interval. Midpoint Method: Approximates using the slope at the midpoint of the interval.

<math-inline>\Delta y = f(x_i, y_i) \cdot \Delta x

<math-inline>y_{i+1} = y_i + f(x_i, y_i) \cdot \Delta x

<math-inline>\frac{dy}{dx} = f(x, y)

<math-inline>| \text{Approximate Value} - \text{Exact Value} |

<math-inline>x_{i+1} = x_i + h

<math-inline>f(x_i, y_i) = \frac{dy}{dx} |_{(x_i, y_i)}

<math-inline>h = \frac{x_{final} - x_{initial}}{n}, where n is the number of steps.

<math-inline>L(x) = f(a) + f'(a)(x-a)

<math-inline>y' = f(x, y)

<math-inline>y_{n+1} = y_n + h cdot f(x_n, y_n)