Calculator Inputs
Use the form for equations in the separable style dy/dx = k · X(x) · y^n.
Formula used
General supported form:
dy/dx = k · X(x) · y^n
Separate the variables:
y^(-n) dy = k · X(x) dx
Integrate both sides:
∫ y^(-n) dy = k ∫ X(x) dx + C
When n ≠ 1:
y^(1-n) / (1-n) = k I(x) + C
When n = 1:
ln|y| = k I(x) + C
The initial condition (x0, y0) finds C. The target value uses the selected target x.
How to use this calculator
- Select the x function model that matches your separable equation.
- Enter the multiplier, x exponent or rate, and y exponent.
- Enter the initial condition using x0 and y0.
- Enter the target x value where you want y.
- Choose RK4 steps and precision. More steps usually improve numerical checking.
- Press Calculate. The result appears above the form.
- Use CSV or PDF buttons to save the output.
Example data table
| Equation | Model settings | Initial condition | Target | Expected idea |
|---|---|---|---|---|
| dy/dx = 2xy | k = 2, X = x^1, n = 1 | x0 = 0, y0 = 1 | x = 1 | y = e^(x^2) |
| dy/dx = 3x^2√y | k = 3, X = x^2, n = 0.5 | x0 = 0, y0 = 4 | x = 1 | Use y^(1-n)/(1-n) |
| dy/dx = e^x y^2 | k = 1, X = e^(1x), n = 2 | x0 = 0, y0 = 0.5 | x = 0.7 | Watch for branch limits |
About this separation method
A separable differential equation can be written as a product of one function of x and one function of y. The calculator uses that structure to move all y terms to one side. It then moves all x terms to the other side. This creates two integrals that can be compared through a constant.
Why this calculator is useful
Manual solving is simple in idea. It can still become error prone. Powers, logarithms, initial values, and negative domains need care. This tool keeps each step visible. It shows the separated statement, the integrated relation, the constant, and the target value. It also compares the closed form result with a Runge Kutta estimate.
Advanced learning value
The numerical table helps you check the curve at many points. The graph shows how the solution changes from the initial point to the target point. When both analytic and numerical values are real, the error column shows agreement. Large errors can mean a singular point, a large step size, or an invalid branch.
Good practice
Start with a known textbook example. Use small steps first. Check whether x or y crosses a forbidden value. For power models, x equals zero can be a problem when the exponent is negative. For fractional powers of y, negative values may leave the real number system. Use the warnings before trusting any final value.
Interpreting results
The constant comes from the initial condition. The target value comes from substituting the target x value into the solved relation. The table provides a practical path between both points. The export buttons save your results for reports, assignments, or later review.
When separation applies
The method applies only when the derivative can be arranged as X(x) times Y(y). Some equations look similar, but they are not separable after simplification. Always isolate dy/dx first. Then test whether every x term can leave the y side. This calculator uses supported models so the algebra remains transparent, repeatable, and easy to verify.
Accuracy notes
Increase the step count when the curve changes quickly. Smaller steps improve the numerical comparison and usually reduce error for stronger study checks.
FAQs
1. What is a separable differential equation?
It is an equation where dy/dx can be written as X(x) times Y(y). This form lets you move y terms with dy and x terms with dx before integrating both sides.
2. Which equation forms does this tool support?
It supports dy/dx = k · X(x) · y^n. X(x) can be a power, exponential, sine, or cosine model. These cover many classroom and applied examples.
3. Why do I need an initial condition?
The initial condition finds the integration constant C. Without x0 and y0, the calculator can show a general relation, but it cannot compute a unique target value.
4. What does the RK4 result mean?
RK4 is a numerical estimate of the same solution. It helps compare the analytic result against a step-based method. Small error usually confirms consistent inputs.
5. Why does the answer show not real?
The selected powers or path may leave the real number system. Fractional powers of negative values and singular points often cause this warning.
6. How many steps should I use?
Use more steps for fast-changing curves or longer intervals. Forty steps works for many examples. Increase the value when the numerical error looks too large.
7. Can this solve every differential equation?
No. It solves supported separable forms only. Linear, exact, homogeneous, Bernoulli, or non-separable equations require different methods or extra transformations.
8. What can I export?
You can export the equation summary, key result values, and numerical table. CSV is useful for spreadsheets. PDF is useful for reports or homework records.