Configure a separable differential equation
Worked examples for quick validation
| Equation | Initial condition | Implicit solution | Example evaluation |
|---|---|---|---|
| dy/dx = 2xy | y(0) = 3 | ln|y| = x² + ln(3) | y(1) ≈ 8.15485 |
| dy/dx = (1/(x+1))(1+0.5y) | y(0) = 2 | 2ln|1+0.5y| = ln|x+1| + 2ln(2) | y(3) = 6 |
| dy/dx = 0.5y(1-y/10) | y(0) = 2 | ln|y/(10-y)| = 0.5x + ln(0.25) | y(4) ≈ 6.48786 |
| dy/dx = sin(x)y² | y(0) = 1 | -1/y = -cos(x) | y(1) ≈ 1.85082 |
Core separable-equation method
For a separable model, write the differential equation as dy/dx = A·X(x)·Y(y). Then move every y-term beside dy and every x-term beside dx.
dy / Y(y) = A·X(x) dx
Integrate both sides to obtain an implicit solution.
∫ dy / Y(y) = A ∫ X(x) dx + C
If an initial condition y(x₀) = y₀ is known, compute C from the integrated relation. Then substitute any target x-value to estimate y on the chosen branch.
Use the calculator step by step
- Choose an x-factor and a y-factor that match your separable differential equation.
- Enter amplitude and any needed exponents, rates, shifts, or carrying capacity values.
- Enable the initial-condition option if you want a specific constant and a numerical y(x) estimate.
- Enter x₀, y₀, and a target x-value for evaluation.
- Press Solve Equation to display the result section above the form.
- Download the report as CSV or PDF when you need to share or archive the work.
Frequently asked questions
1. What makes an equation separable?
A separable equation can be rewritten so one side contains only y with dy, while the other side contains only x with dx. After that, both sides are integrated independently.
2. Why does the calculator return an implicit solution?
Many separable differential equations integrate cleanly but do not isolate y in a simple explicit form. An implicit solution still captures the relationship exactly and often supports numerical evaluation.
3. Why do domain warnings appear?
Warnings appear when logarithms, reciprocal factors, or restricted branches are involved. They help you avoid singular points, undefined terms, and real-valued branch changes in the integrated solution.
4. Can I solve logistic-type equations here?
Yes. Select the logistic y-factor y(1-y/K). The calculator integrates the standard partial-fraction form and can estimate y(x) on the principal branch between 0 and K.
5. What does the initial slope tell me?
The initial slope is dy/dx evaluated at the supplied starting point. It shows whether the solution is increasing, decreasing, or momentarily flat at that specific condition.
6. Why can a target evaluation fail?
Evaluation can fail when the chosen branch becomes non-real, crosses a singularity, or violates logarithmic or exponential restrictions. In that case, the implicit solution remains useful even without a real-valued estimate.
7. Is this calculator suitable for homework checking?
Yes. It is useful for checking algebraic separation, confirming constants from initial conditions, and validating quick numerical evaluations. You should still show your derivation when submitting formal work.
8. Does the PDF include the result summary?
Yes. The PDF button exports the solved result panel shown above the form. Solve the equation first, then download the summary for revision notes or documentation.