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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.

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Open Second-Order ODE Solver and you get an answer straight away, with no account to create. It covers undamped, under-damped, critically damped, over-damped — 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 is 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.

What does it take into account?

It factors in undamped, under-damped, critically damped, over-damped. Change any of them and the output follows immediately.

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

They sit next to each other but answer different questions. 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. 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.