Systems of Differential Equations

A system of differential equations is a set of two or more differential equations involving the same variables.

General Form

The general form of a system of n first-order differential equations is:

dx₁/dt = f₁(t, x₁, x₂, ..., xₙ)
dx₂/dt = f₂(t, x₁, x₂, ..., xₙ)
...
dxₙ/dt = fₙ(t, x₁, x₂, ..., xₙ)

where f₁, f₂, ..., fₙ are given functions.

Solving Systems of Differential Equations

The methods for solving a system of differential equations depend on the specifics of the system. Here are a few general methods:

  1. Elimination: If one equation can be algebraically manipulated to isolate one variable, substitution can be used to eliminate that variable from the other equation(s), resulting in a system with one fewer equation and one fewer variable. This process can be repeated until a single differential equation in one variable is obtained, which can then be solved by standard methods.

  2. Matrix Methods: For a system of linear differential equations, it can be rewritten in matrix form as X' = AX where X is the vector of variables and A is the matrix of coefficients. If the matrix A can be diagonalized, the system can be solved by decoupling the equations.

  3. Numerical Methods: For complex systems, or systems that cannot be solved analytically, numerical methods such as Euler's method, the Runge-Kutta method, or software-based solvers can be used to approximate the solution.

Applications

Systems of differential equations arise in many fields whenever multiple quantities change simultaneously and affect each other. This includes areas such as physics (e.g., modeling interacting particles or predator-prey relations in population dynamics), engineering (e.g., analyzing interconnected circuits), and economics (e.g., modeling multiple interacting markets).