It uses the tangent line at a known point to estimate the function's value at a nearby point, then repeats this process.

It approximates the curve with a series of line segments, which deviates from the true solution.

Smaller step sizes generally lead to more accurate approximations, but require more calculations.

The initial condition provides the starting point (x0, y0) for the approximation process.

Problems where an analytical solution to the differential equation is difficult or impossible to obtain.

It approximates the solution curve by following tangent lines at discrete points.

The derivative gives the slope of the tangent line, which is used to estimate the next point.

It can be inaccurate, especially with larger step sizes, and may diverge from the true solution.

It's the error introduced in a single step of the method, due to approximating the curve with a line.

Use a smaller step size or a higher-order numerical method.

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)