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First-Order ODE Solver

Solves dy/dx = f(x, y) from an initial condition with RK4, integrating outward in both directions, and plots the solution curve. Growth, decay, Newton cooling, logistic and separable cases are one click away.

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First-Order ODE Solver works straight from this page — free, instant, nothing to install. You will find it under Calculus, with Second-Order ODE Solver and Bernoulli ODE Solver for the neighbouring cases.

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 is First-Order ODE Solver?

Solves dy/dx = f(x, y) from an initial condition with RK4, integrating outward in both directions, and plots the solution curve. Growth, decay, Newton cooling, logistic and separable cases are one click away.

When would I actually use this?

Checking a derivative or an integral you worked out by hand, finding where a function turns, and getting a numeric answer when no closed form exists.

What is the most common mistake?

Integrating across a discontinuity as if it were not there. A numeric method will happily return a finite value for an integral that diverges — check the domain before trusting the number.

How is First-Order ODE Solver different from Second-Order ODE Solver?

They sit next to each other but answer different questions. Second-Order ODE Solver: solves a·y″ + b·y′ + c·y = g(x) with two initial conditions. Gives the characteristic roots, names the regime (undamped, under-damped, critically damped, over-damped), derives a particular solution for constant, polynomial and exponential forcing, and plots the result. Pick whichever matches what you're starting from — both are free.

Is there a tool for the next step?

Bernoulli ODE Solver is the closest one after this: Solves y′ + P(x)y = Q(x)yⁿ by showing the substitution v = y^(1−n) that turns it into a linear equation, then integrating. The two degenerate cases are handled explicitly: n = 0 is already linear, n = 1 is separable.

What else is worth having open alongside it?

System of ODEs Solver and Related rates solver — they come up in the same task often enough to be worth a second tab.

Where do the figures come from?

Symbolic results are exact; numeric ones come from adaptive quadrature or a standard step method, and the tool reports which. Where both are available, compare them — a large gap means the problem is ill-conditioned.

Further reading

All guides →
ExplainerOrder of Operations (PEMDAS / BODMAS): Why the Order MattersPEMDAS and BODMAS are two names for one rule that fixes the order you evaluate a math expression — so everyone gets the same answer. Learn the rule with a worked example.ExplainerCombinations vs Permutations: When Order MattersThe difference between combinations and permutations comes down to one question: does order matter? Learn the nCr and nPr formulas with clear worked examples.How-toHow Much Laundry Detergent to Use: Load Size First, Water Hardness SecondA normal medium load in a high-efficiency machine works out at 37 ml. Load size swings that threefold; water hardness only 1.28-fold — and the tool's hardness input carries no units at all. Here are the scales and the conversions it leaves out.ExplainerThe Pigeonhole Principle Proves a Thing Exists Without Finding ItTen items in three containers force four into one of them. 367 people force a shared birthday. Neither statement tells you which container or which pair.ExplainerA Cone Is Exactly One Third of Its CylinderRadius 5, height 10: the cone holds 261.8 and the cylinder around it holds 785.4. The ratio is exactly a third, and it holds for any base shape at all.ExplainerOne Formula That Is Also a Rectangle and Also a TriangleParallel sides of 8 and 5 with a height of 4 give an area of 26 — the average width times the height. Make the two sides equal and it is a rectangle; shrink one to nothing and it is a triangle.