Calculators / Differential Equations / 2×2 Linear System ODE Solver

2×2 Linear System ODE Solver

Published on: December 14, 2025

This 2×2 Linear System ODE Solver helps you solve a system of two 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' = ...
  2. Rewrite the system in matrix form X' = A X.
  3. Compute eigenvalues from det(A − λI) = 0.
  4. Compute eigenvectors for each eigenvalue.
  5. Write the general solution using eigenpairs.
  6. If initial conditions are given, solve for constants.
  7. Classify the equilibrium using eigenvalues.

Formula bank:

Matrix form: X′ = A X
det(A − λI) = 0
X(t) = C1v1eλ1t + C2v2eλ2t
If λ = α ± iβ: X(t) = eαt(C1U(t) + C2V(t))