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System of ODEs Solver

Solves 2×2 and 3×3 linear systems x′ = Ax with eigenvalues, eigenvectors and the closed-form general solution, classifies the fixed point (node, spiral, centre, saddle, degenerate), and integrates arbitrary nonlinear 2D systems numerically with a phase portrait.

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Open System of ODEs Solver and you get an answer straight away, with no account to create. It covers node, spiral, centre, saddle, degenerate — adjust any of them and the result follows immediately.

How to use it

  1. Open the tool — no signup or install needed.
  2. Enter your input or adjust the available options.
  3. Get your result instantly, then copy or download it.

Frequently asked questions

What does System of ODEs Solver do?

Solves 2×2 and 3×3 linear systems x′ = Ax with eigenvalues, eigenvectors and the closed-form general solution, classifies the fixed point (node, spiral, centre, saddle, degenerate), and integrates arbitrary nonlinear 2D systems numerically with a phase portrait.

What does it take into account?

It factors in node, spiral, centre, saddle, degenerate. Change any of them and the output follows immediately.

When would I actually use this?

Getting through a problem set: solving for the unknown, factoring an expression, and — more usefully — seeing the steps that got there.

What is the most common mistake?

Losing a solution when both sides are squared or divided by an expression. Squaring can add roots that do not satisfy the original equation, and dividing can remove one — check every answer back in the original.

How is System of ODEs Solver different from System of Linear Equations Solver?

They sit next to each other but answer different questions: System of Linear Equations Solver is the one to open when you need it to solve 2×2, 3×3 and 4×4 linear systems by Gaussian elimination, Cramer's rule or the matrix inverse — with the determinant, step-by-step working, and detection of no-solution and infinite-solution cases. Pick whichever matches what you're starting from — both are free.

Is there a tool for the next step?

System of Nonlinear Equations Solver is the closest one after this: Solve 2- or 3-variable nonlinear systems (polynomial, trigonometric, exponential) with multivariate Newton's method — enter equations as text, set a starting guess, and surface every real solution with its residual.

What else is worth having open alongside it?

System of Inequalities Grapher and Absolute Value Equation Solver — they come up in the same task often enough to be worth a second tab.

Where do the figures come from?

The algebra is symbolic where the expression allows it and numeric otherwise, and the tool says which it used. Exact results are given as fractions or radicals rather than rounded decimals.

Further reading

All guides →
How-toHow to Use the Quadratic Formula: Solving ax² + bx + c = 0Solve any quadratic equation with x = (−b ± √(b² − 4ac)) / 2a, and read the discriminant to know how many real solutions to expect.How-toHow to Find a Cube Root by HandA step-by-step guide to finding a cube root by estimation and refinement — plus how it differs from a square root and when a value stays irrational.ExplainerHow Exponents Work: Powers and the Laws of ExponentsAn exponent is repeated multiplication. Learn the laws of exponents, what negative and fractional powers mean, and why anything to the zero is one.ExplainerThe Flat Pattern of a Cone Is a Sector, Not a CircleRoll a 50 mm cone 80 mm tall and its pattern is a 190.8° slice of a 94 mm circle. Flatten a nearly cylindrical lampshade and the pattern radius runs to 839 mm — the shallower the taper, the further away the apex.ExplainerA 1 % Drop Rate Does Not Mean One Hundred RunsAt 1 %, a hundred attempts give you 63.4 % — not certainty. Ninety per cent takes 230 attempts and ninety-nine takes 459, and more than a third of players are still empty-handed at a hundred.ExplainerFibonacci and the Golden RatioThe Fibonacci sequence adds each pair of terms to make the next; the ratio of neighboring terms homes in on the golden ratio φ ≈ 1.618. See how, and where the pattern turns up.