Calculator input
Example data table
| Item | Example value | Meaning |
|---|---|---|
| Order | 3 | Third-order initial value problem. |
| Equation | 1·y''' + 2·y'' + 0·y' + 1·y = sin(x) | Constant-coefficient model with sinusoidal forcing. |
| Initial conditions | y(0)=0, y'(0)=1, y''(0)=0 | State values at the start point. |
| Domain | x from 0 to 10 | Integration interval. |
| Step size | 0.1 | Uniform RK4 marching increment. |
Formula used
The solver handles the general constant-coefficient model aₙy^(n) + aₙ₋₁y^(n-1) + ... + a₁y' + a₀y = g(x). It converts the nth-order equation into a first-order system.
Each step then uses classical fourth-order Runge-Kutta integration:
RK4 is accurate for smooth problems, but very stiff systems may still require smaller steps or specialized stiff solvers.
How to use this calculator
- Select the order of the differential equation.
- Enter the interval start, interval end, and step size.
- Provide all coefficients from a₀ through aₙ.
- Enter the required initial conditions at x₀.
- Choose a forcing model and fill its active parameters.
- Press Solve equation to view the response, graph, and downloadable results.
FAQs
1) What kind of equations does this solver handle?
It solves linear higher-order ordinary differential equations with constant coefficients and common forcing terms. The method converts the equation into a first-order system and integrates the system numerically.
2) Why are several initial conditions required?
An nth-order problem needs n starting values. For example, a third-order equation requires y, y', and y'' at the same starting x-value.
3) What does the step size control?
The step size sets the spacing between numerical evaluation points. Smaller steps often improve accuracy, but they also increase runtime and enlarge the output table.
4) Can I solve backward in x?
Yes. If the end value is less than the start value, the solver automatically marches backward while preserving the entered positive step magnitude.
5) Does the graph show the forcing too?
Yes. The Plotly chart compares y(x) against g(x), which helps you see whether the response tracks, lags, or amplifies the applied excitation.
6) When should I reduce the step size?
Reduce the step if the solution changes sharply, oscillates unnaturally, or behaves very differently after small parameter changes. Step sensitivity is a practical warning sign.
7) Is this better than a symbolic solver?
Not always. Symbolic solvers can produce exact formulas, but numerical solvers are faster for experimentation, parameter sweeps, and problems where closed-form expressions are inconvenient.
8) What do the CSV and PDF downloads contain?
The CSV export includes the full numerical grid. The PDF export summarizes the setup and adds a formatted results table for convenient reporting or sharing.