Online Tool Store
Differential Equation Solver
Solve first-order and second-order differential equations numerically with RK4, a live chart, a step-by-step value table, and CSV export.
This works right here on your device — nothing gets sent out.
Supports + − × ÷ ^, parentheses, sin, cos, tan, sqrt, exp, ln, abs, and the constants pi and e. Use x and y.
How Differential Equation Solver works
- Pick 1st order (dy/dx = f(x, y)) or 2nd order (y'' = f(x, y, y')), or load a preset to see a worked example.
- Type the right-hand side of the equation, then set the starting point (x₀, y and, for 2nd order, y') and how far to solve along x.
- The solver steps forward using the fourth-order Runge-Kutta (RK4) method and plots y(x) — and y'(x) in 2nd-order mode — on the chart.
- Scroll the step table to see the value at every computed point, or download the full table as a CSV.
The method
RK4 estimates each step by sampling the equation's slope at four points across the interval and
combining them with a weighted average, which is far more accurate per step than simply following
the slope at the start of the interval (Euler's method). For a step size h:
y(x + h) ≈ y(x) + h/6 · (k₁ + 2k₂ + 2k₃ + k₄)
where k₁ through k₄ are the slope estimates at the start, two midpoints, and the end of the interval. 2nd-order equations are solved the same way by treating y' as a second unknown alongside y.
FAQ
Does this give an exact formula, or a numeric answer?
It's numeric, not symbolic — it uses the classic fourth-order Runge-Kutta (RK4) method to step the solution forward from your starting point, the same technique taught in any numerical methods course. For equations with a known closed-form solution (like exponential growth or a simple oscillator), RK4's result matches the exact formula to about 8-10 significant figures at the step counts used here.
What equations can I type in?
Any combination of + − × ÷ ^, parentheses, sin, cos, tan, sqrt, exp, ln, log, abs, and the constants pi and e, using x and y (and v for y' in 2nd-order mode). For example: 0.3*y*(1-y/10), or sin(x)-y.
What's the difference between 1st order and 2nd order mode?
1st order solves dy/dx = f(x, y) from a single starting point (x0, y0) — growth, decay, and similar rate equations. 2nd order solves y'' = f(x, y, y') from a starting position and starting velocity (x0, y0, y'(x0)) — oscillators, springs, and anything with acceleration.
Why did it say the solution diverges?
Some equations grow without bound in finite (or near-finite) x — the numbers overflow before reaching the end of your chosen x range. Try a shorter range, a different starting value, or fewer steps.
Does anything I type get sent anywhere?
No. The equation is parsed and solved entirely in your browser — nothing is uploaded.
How we compare
| Feature | Online Tool Store | Symbolab | MathDF |
|---|---|---|---|
| Free-form f(x, y) input, not limited to named equation forms | Included | Limited to recognized patterns | Limited to recognized patterns |
| 2nd-order (oscillator/spring) equations | Included | Included | Included |
| Runs fully in your browser, nothing uploaded | Included | Server-side solving | Server-side solving |
| Chart of the solution as it solves | Included | Included | Included |
| Step-by-step numeric value table with CSV export | Included | Not included | Values table, no export |
| Symbolic step-by-step algebra (CAS) | Not included | Included | Included |
Choose Symbolab or MathDF when you need the symbolic algebra steps for a textbook-style closed-form answer. Choose Differential Equation Solver when you want a fast numeric solution and chart for any equation you can type — including ones without a tidy closed form — with the full step table exportable as CSV.