Differential Equation Calculator — Solve Any ODE or PDE Free, With Steps

DiffCalc is a free, browser-based differential equation solver built on a symbolic CAS engine (mathjs + Algebrite). Type any equation — separable, linear, exact, Bernoulli, Riccati, Cauchy–Euler, second-order constant-coefficient, Laplace, systems, or PDE — and get the exact symbolic solution, a step-by-step derivation, the classification, and an interactive plot. No sign-up, no AI guesses.

What you can solve

First-order ODEs (separable, linear with integrating factor, exact, homogeneous, Bernoulli, Riccati, Lagrange, Clairaut), second-order linear ODEs (constant coefficients, Cauchy–Euler, reduction of order, undetermined coefficients, variation of parameters), higher-order constant-coefficient equations, systems of ODEs (eigenvalue method, matrix exponential, RK4 numerical fallback), Laplace transforms and inverse Laplace with initial value problems, power-series and Frobenius solutions, Wronskian computations, and separation-of-variables PDEs (heat, wave, Laplace).

How it works

  1. Classify — the engine detects order, degree, linearity, and ODE/PDE type.
  2. Route — the right specialized solver runs (separable, IF, Laplace, eigenvalue, etc.).
  3. Solve symbolically — exact closed-form output when one exists; numerical RK45 when it does not.
  4. Verify — the candidate solution is substituted back; you see PASS / UNVERIFIED / FAIL honestly.
  5. Explain — every step cites the rule used, with a worked example and a plot.

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Why DiffCalc

Most online ODE tools either wrap an AI that hallucinates wrong solutions, or charge for steps. DiffCalc runs a real symbolic engine in your browser, returns exact answers, and shows every algebraic move. It is built and maintained by Salman Ali, with a published methodology and an about page describing how each solver is tested against Boyce & DiPrima and Zill reference problems.