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
- Euler value at endpoint
- 0.25
- Value with half the step
- 0.375
- Absolute step-halving discrepancy
- 0.125
- Coarse solution trace: x,y
- 0, 0; 0.5, 0; 1, 0.25
Try another scenario
- Restricted derivative y′=f(x,y)
- y
- Initial x
- 0
- Initial y
- 1
- Endpoint x
- 1
- Positive coarse step
- 0.5
- Euler value at endpoint
- 2.25
- Value with half the step
- 2.441406
- Absolute step-halving discrepancy
- 0.191406
- Coarse solution trace: x,y
- 0, 1; 0.5, 1.5; 1, 2.25
Calculation method
Calculation method
Euler value at endpoint
Advance y←y+h×f(x,y), shortening only the last step to hit the endpoint.
Value with half the step
Repeat using half the entered step.
Absolute step-halving discrepancy
Absolute difference between coarse and refined endpoint values; not a guaranteed error bound.
Coarse solution trace: x,y
Include the initial condition 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
- A scalar first-order initial-value problem integrated forward with at most 1,000 coarse steps. Explicit Euler can be unstable, especially for stiff equations; discrepancy alone does not establish accuracy. 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.