Ordinary Differential Equation Calculator

What does this ordinary differential equation calculator compute?

A scalar first-order initial-value problem y′=f(t,y), y(t0)=y0 is stepped to t1. This calculator uses forward Euler or classical RK4. Linear rhs is y′=k y with exact y0 e^{k(t1−t0)}. Polynomial rhs is y′=p(t) with exact y0+∫p.

The published formula or method is Euler / classical RK4. The following references support the stated mathematics or method:

• Gilbert Strang, Calculus — 16.2 Differential Equations: Forward Euler and classical four-stage fourth-order Runge–Kutta for first-order initial-value problems, together with step-size accuracy and stability discussion. Mathematical reference only; not validation or endorsement of this implementation.

Which inputs and limits should you check?

Step size and stability matter. Successful integration is not a blanket accuracy guarantee; retain the method, step count, diagnostics, and complete trajectory.

The worked example uses rk4, n=40, k=1 with y = e. Compare it with the Example calculations and the complete result for the same inputs.

How can you verify and cite this result?

CalculatorX is responsible for this page. The mathematical references above explain the model; implementation checks and their scope are documented separately in the math.numerical.ode v1.1.1 Evidence and capability contract. Read the displayed verification and production state together with the specification's numerical limits.

Use the clean calculator URL to cite the tool. A shared input link identifies a particular calculation. Content updates, numerical verification, and domain reviews have separate dates and scopes.

How are precision, sharing and exports handled?

Euler and classical RK4 complete the requested fixed step count; this is not an adaptive accuracy acceptance. Compare the numerical result with the closed-form reference and inspect stability warnings. abs_err is a legacy signed numerical-minus-reference difference before12-digit publication rounding; diagnostics.absolute_error is its magnitude. Neither is a certified error bound. Inputs, coefficients and h retain binary64 precision; numerical and reference endpoints publish12 significant digits.

Polynomial degree0–8 is supported, with constant-first coefficients (a constant can be 2). Empty coefficient tokens are rejected. Exact dyadic intermediate evaluation and composite quadrature avoid polynomial cancellation and stage overflow; final endpoint/reference values must fit binary64. A nonzero polynomial endpoint/reference that projects to zero is refused. Linear fixed-step zero can occur through cancellation or underflow and is explicitly warned when the analytic reference is nonzero. The graph shows at most257 samples of the discrete trajectory; JSON/CSV contain the complete endpoint result, model inputs, diagnostics and warnings, not every step.

Ordinary editing computes locally. Opening a share URL sends its query inputs to the server, and inputs may appear in server or third-party logs. REST/MCP send inputs to the server. CSV/JSON downloads and clipboard actions are explicit; local calculation history may store results on this device. Clear removes the current form values; use the history controls to remove retained history. REST/MCP version pins request the supported current calculation contract. Ordinary page links restore inputs into the current engine and show a notice when the linked version differs; neither path provides a historical engine service.