Examples
Examples
Try the starting example
- Restricted derivative y′=f(x,y)
- x
- Initial x
- 0
- Initial y
- 0
- Endpoint x
- 1
- Positive coarse step
- 0.5
- Fourth-order RK value at endpoint
- 0.5
- Value with half the step
- 0.5
- Step-halving discrepancy divided by 15
- 0
- Coarse solution trace: x,y
- 0, 0; 0.5, 0.125; 1, 0.5
Try another scenario
- Restricted derivative y′=f(x,y)
- y
- Initial x
- 0
- Initial y
- 1
- Endpoint x
- 1
- Positive coarse step
- 0.5
- Fourth-order RK value at endpoint
- 2.717346
- Value with half the step
- 2.71821
- Step-halving discrepancy divided by 15
- 0.000058
- Coarse solution trace: x,y
- 0, 1; 0.5, 1.6484375; 1, 2.71734619141
Calculation method
Calculation method
Fourth-order RK value at endpoint
Classical RK4 uses four derivative samples per step and weights (1,2,2,1)/6, shortening the final step to hit the endpoint.
Value with half the step
Repeat with half the entered step.
Step-halving discrepancy divided by 15
|fine−coarse|/15 is an order-four smooth-problem diagnostic, not a certified error bound.
Coarse solution trace: x,y
Include initial x,y and every coarse step.
Symbols and units
- a
- Restricted derivative y′=f(x,y)
- b
- Initial x
- c
- Initial y
- d
- Endpoint x
- e
- Positive coarse step
Use the units shown. A list keeps the order in which you enter its values. The symbol ^ means “raised to the power of”.
Assumptions & limits
Assumptions & limits
- Forward scalar first-order IVP with at most 1,000 coarse steps. Classical explicit RK4 is not a stiff solver; discontinuities, blow-ups and rapidly varying behaviour need separate treatment. Restricted expressions allow x (and y for ODEs), pi, e, + - * / ^, and sin, cos, tan, sec, csc, cot, exp, log, sqrt, abs, sinh, cosh, tanh, sech, csch, coth. Trigonometric arguments are in radians. Double-precision results are approximate. Numeric tables use 12 significant digits and display components at roundoff scale (32 machine epsilons relative to their row or relevant calculation scale) as zero; scalar diagnostics retain full precision.
Are these inputs private?
These calculations run in your browser. A shared link contains the values you choose to share. Only questions submitted to the separate AI finder are sent to OpenAI.
Further reading
SciPy: numerical integration and ODE method references ↗
Background reading; this does not imply endorsement or independent review of SumFizz.