Systems of First Order Linear Ordinary Differential Equations with Constant Coefficients
engineering fundamentals Systems of First Order Linear
ODE with Constant Coefficients

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Standard Form

A system of first order linear ordinary differential equation can be expressed as the following form

or in the matrix form

where the matrix contains only constants and is function of .

One way to solve these n coupled first order linear ODE is to diagonalize the coefficient matrix and thus decouple these equations.

Suppose that and are distinct eigenvalues and associated eigenvectors of

Let and


the system of differential equations can be rewritten as

Move to the right hand side of the equation, we have

where

and

The general solution of this system of equations

is

Substituting back to , the original system of differential equations can be solved:

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