Calculators / Differential Equations / 3×3 Linear System ODE Solver

3×3 Linear System ODE Solver

Published on: December 21, 2025

This 3×3 Linear System ODE Solver helps you solve a system of three first-order differential equations and shows each step clearly. It can be used to work with coupled linear systems, apply suitable solution methods, and optionally use initial conditions to find a particular solution. This makes it useful for checking answers, understanding how linear systems of ODEs are solved, and practising differential equations step by step.

Step-by-step method

  1. Write the system clearly: x' = ..., y' = ..., z' = ...
  2. Rewrite the system in matrix form X' = A X.
  3. Compute eigenvalues from det(A − λI) = 0 (or use matrix exponential).
  4. Compute eigenvectors (or a real basis when complex eigenvalues occur).
  5. Write the general solution.
  6. If initial conditions are given, solve for constants.

Formula bank:

Matrix form: X′ = A X
det(A − λI) = 0
If A has 3 independent eigenvectors: X(t) = Σ Ckvkeλkt
Always: X(t) = eAtC