SciPy is a Python library that provides a comprehensive set of tools for scientific computing, including a module dedicated to solving ODEs.

Runge-Kutta methods are a family of explicit iterative methods for solving ODEs, known for their accuracy and stability.

RK4, also known as the classical Runge-Kutta method, is a fourth-order Runge-Kutta method commonly used for solving first-order ODEs due to its balance of accuracy and computational efficiency.

The RK4 method is a fourth-order method, meaning that the local truncation error is proportional to $h^5$, where $h$ is the step size.

Explicit Runge-Kutta methods can suffer from stability issues, especially for stiff ODEs, where the solution contains rapidly varying components.

Implicit Runge-Kutta methods are generally more stable than explicit methods, making them suitable for solving stiff ODEs.

RK6, also known as the Radau IIA method, is a sixth-order implicit Runge-Kutta method commonly used for solving stiff ODEs.

The RK6 method is a sixth-order method, meaning that the local truncation error is proportional to $h^7$, where $h$ is the step size.

Implicit Runge-Kutta methods are generally more stable than explicit methods, making them suitable for solving stiff ODEs.

Implicit Runge-Kutta methods are generally more computationally expensive than explicit methods due to the need to solve a system of nonlinear equations at each step.

Adaptive step size control adjusts the step size based on the local error estimate, improving accuracy, reducing computational cost, and enhancing stability.

The Runge-Kutta-Fehlberg method is a commonly used adaptive step size control algorithm that combines a fourth-order RK method with a fifth-order RK method to estimate the local error.

The Runge-Kutta-Fehlberg method is a fourth-order method, meaning that the local truncation error is proportional to $h^5$, where $h$ is the step size.

Adaptive step size control can increase computational cost due to the need to evaluate the local error estimate at each step.

In implicit Runge-Kutta methods, the Jacobian matrix is used to solve the system of nonlinear equations that arise at each step.