Calculated result
Review the solution, errors, graph, and verification details.
Formula used
Euler uses the current slope for each step. Runge–Kutta combines several slopes for better accuracy. Adaptive methods change step sizes when errors increase significantly.
Euler: yₙ₊₁ = yₙ + h f(xₙ, yₙ) RK4: yₙ₊₁ = yₙ + h(k₁ + 2k₂ + 2k₃ + k₄)/6
How to use
Select the equation type and enter derivative functions. Supply every required initial condition and integration endpoint. Choose a method, then inspect results and warnings carefully.
- Choose first-order, second-order, or coupled system mode.
- Enter derivative expressions using the supported function syntax.
- Provide initial values, interval settings, and numerical controls.
- Optionally enter an exact solution for error comparison.
- Submit the form and review verification information.
Example data
| Model | Derivative function | Initial data | Suggested interval |
|---|---|---|---|
| Growth | 0.5*y | y(0)=1 | 0 to 5 |
| Logistic | 0.8*y*(1-y/100) | y(0)=5 | 0 to 12 |
| Oscillator | -4*y | y(0)=1, y′(0)=0 | 0 to 10 |
| Predator-prey | 1.1*y-0.4*y*z | y(0)=10, z(0)=5 | 0 to 20 |
Important limitations
The symbolic recogniser covers common educational equation patterns. General nonlinear symbolic solving requires specialised computer algebra. Numerical results still require sensible intervals and step controls.
Frequently asked questions
What is an initial value problem?
It combines a differential equation with starting values. Those values determine a particular solution from possible solutions. They also anchor numerical integration at one known point.
Which method should I choose?
RK4 works well for many smooth nonstiff problems. Euler helps demonstrate basic numerical integration behaviour clearly. Adaptive RKF45 suits changing slopes and accuracy requirements.
Why does step size matter?
Large steps finish faster but may reduce accuracy. Small steps improve detail but increase computation and storage. Compare methods before trusting a difficult solution completely.
Can this solve second-order equations?
Yes, enter the expression for the second derivative. The calculator converts it into two first-order equations. Provide both position and first-derivative initial values.
Can it solve coupled systems?
Yes, system mode supports two dependent state variables. Enter separate derivative functions for y and z. The phase portrait displays their joint numerical trajectory.
How are exact errors calculated?
Enter a valid exact expression for y. The calculator evaluates it at every numerical point. Absolute and relative errors then appear within tables.
What indicates an unstable result?
Rapid growth and non-finite values can signal instability. Excessive step sizes may also create false oscillations. Reduce step size and compare reliable methods again.
Does symbolic recognition handle every equation?
No, it recognises several common educational patterns only. Unrecognised equations still receive numerical solutions and classifications. Use specialist algebra systems for comprehensive symbolic solving.
Can results be exported?
Yes, tables can download as CSV files. Reports can download as PDF documents too. Graphs and LaTeX output support further reporting work.