Model linked rate equations with stepwise solving and outputs. Track variables across custom intervals easily. Download polished tables for classwork, checks, documentation, and sharing.
| Scenario | dy1/dx | dy2/dx | Initial State | x Range | Step |
|---|---|---|---|---|---|
| Simple harmonic pair | y2 | -y1 | y1(0)=1, y2(0)=0 | 0 to 6.28 | 0.1 |
| Coupled growth and decay | 0.5*y1 - 0.2*y2 | 0.3*y1 - 0.4*y2 | y1(0)=2, y2(0)=1 | 0 to 5 | 0.05 |
| Three-state system | y2 | y3 | y1(0)=1, y2(0)=0, y3(0)=-1 | 0 to 4 | 0.1 |
This calculator applies the classical fourth-order Runge-Kutta method to a first-order system. For each dependent variable, the next value is estimated using four slope checks inside every step.
For a system written as dY/dx = F(x, Y), the update is:
Y(n+1) = Y(n) + (h/6) × (k1 + 2k2 + 2k3 + k4)
Where:
k1 = F(xn, Yn)
k2 = F(xn + h/2, Yn + h·k1/2)
k3 = F(xn + h/2, Yn + h·k2/2)
k4 = F(xn + h, Yn + h·k3)
This method balances accuracy and speed. It works well for many educational, engineering, and scientific systems when the step size is chosen carefully.
Choose whether your system has two or three equations.
Enter the start value of x, end value of x, and the numerical step size.
Provide the initial values for y1, y2, and optionally y3.
Type each differential equation using x, y1, y2, and y3.
Press the solve button to compute the solution table.
Review the summary cards, plotted curves, and full output table.
Use the CSV button for spreadsheet work and the PDF button for a print-ready export.
This tool solves coupled first-order ordinary differential equations numerically. It is useful when exact symbolic solutions are difficult, unavailable, or unnecessary. Students, teachers, analysts, and researchers can test system behavior over selected intervals and compare variable trends step by step.
Many real systems are coupled. Population models, electrical circuits, chemical balances, motion equations, and control systems often depend on several changing quantities at once. A numerical method lets you estimate values across time or distance without deriving a closed-form expression for every case.
The fourth-order Runge-Kutta method is widely used because it gives strong accuracy for many practical problems. It improves on simple Euler stepping by checking intermediate slopes. That reduces accumulated error and produces smoother tables and graphs for ordinary system analysis.
It solves coupled first-order ordinary differential equations entered as dy/dx expressions. You can model two or three linked variables over a chosen x interval.
No. It produces numerical approximations using fourth-order Runge-Kutta steps. That makes it useful for practical estimation, plotting, and verification work.
You may use x, y1, y2, and y3. Common functions such as sin, cos, exp, log, sqrt, and abs are also supported.
A smaller step often improves accuracy but increases calculation length. A larger step runs faster but may miss important behavior or reduce stability.
You can convert a higher-order equation into a first-order system. Then enter the equivalent linked equations using extra variables.
Try a smaller step size, check your equation syntax, and review initial values. Some systems are sensitive and need finer numerical resolution.
The CSV export contains the computed solution rows. Each row includes step index, x, and the solved variable values from the table.
Use the PDF option when you need a printable report for notes, assignments, review meetings, or documentation records.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.