Calculator Input
Choose a separable structure of the form dy/dx = f(x)·g(y). The calculator finds an implicit solution, estimates the target value, and plots a numerical curve.
Plotly Graph
The graph is generated numerically using the chosen initial condition. It complements the implicit symbolic step shown above.
Formula Used
For a separable first-order equation, the model is written as:
Move every y-term to the left and every x-term to the right:
Integrate both sides:
Then apply either an initial condition (x₀, y₀) or a manual constant. This calculator computes the implicit solution, estimates y at the target x, and plots a numerical branch.
How to Use This Calculator
- Select the x-side family f(x).
- Select the y-side family g(y).
- Enter the coefficients, exponents, shifts, and rates.
- Provide the initial condition x₀ and y₀.
- Enter the target x where you want the estimate.
- Set the graph interval and point count.
- Use manual C only when you want a custom branch.
- Press Solve Equation to display the result above the form.
Example Data Table
| Example | f(x) | g(y) | x₀ | y₀ | Target x | Implicit relation | Estimated y(target) |
|---|---|---|---|---|---|---|---|
| Classical growth | 2x | y | 0 | 1 | 1 | ln|y| = x² + C | 2.718282 |
| Reciprocal forcing | 3/(x+1) | (y)^2 | 1 | 2 | 2 | -1/y = 3ln|x+1| + C | 0.511483 |
| Exponential response | e^x | 2e^y | 0 | 0 | 0.5 | -e^-y / 2 = e^x + C | 0.847862 |
Frequently Asked Questions
1) What qualifies an equation as separable?
A separable equation can be rearranged into a product form where all y terms move to one side and all x terms move to the other side before integration.
2) Does this tool produce explicit solutions every time?
No. Many separable equations are easier to express implicitly. This tool always returns the implicit relation and then estimates the target value numerically when needed.
3) Why can the target value become undefined?
Undefined results usually appear near singularities, poles, zero denominators, or branch changes. These features can prevent a real-valued estimate for the chosen condition.
4) What is the role of the integration constant C?
C selects one member of the solution family. The calculator can infer C from an initial condition or accept a manual value for branch exploration.
5) Why does the graph use numerical integration?
Implicit formulas are often difficult to invert cleanly. Numerical plotting provides a practical curve while preserving the symbolic separation and integration steps.
6) Are the trigonometric inputs interpreted in degrees?
No. Trigonometric expressions in this calculator use radians. Enter phase shifts and rates consistently if your model comes from physics or engineering notation.
7) Can I model logistic-style behavior here?
Yes, if the chosen structure can be written as f(x)g(y). Many growth and decay models fit naturally after selecting an appropriate y-family.
8) What do CSV and PDF exports contain?
The exports capture the main summary values, the separated form, the implicit relation, and the target estimate shown in the results panel.