Calculator Inputs
Plotly Graph
Submit the calculator to generate a potential surface and solution level view.
Formula Used
An exact differential equation has the form:
M(x,y) dx + N(x,y) dy = 0
The equation is exact when:
∂M/∂y = ∂N/∂x
Then a potential function exists:
F_x = M and F_y = N
The implicit solution is:
F(x,y) = C
This page estimates the potential with two line-integral paths:
F ≈ C + ∫ M(x,y0) dx + ∫ N(x,y) dy
A small path difference supports exactness. A large difference warns that the equation may not be exact.
How to Use This Calculator
- Enter the
M(x,y)expression besidedx. - Enter the
N(x,y)expression besidedy. - Choose a base point and target point.
- Set tolerance, derivative step, integration steps, and graph limits.
- Press calculate. The result appears above the form.
- Review exactness status, residuals, path difference, and potential estimate.
- Use the CSV or PDF button to save the report.
Example Data Table
| M(x,y) | N(x,y) | dM/dy | dN/dx | Expected Result |
|---|---|---|---|---|
2*x*y + cos(x) |
x^2 + 2*y |
2*x |
2*x |
Exact |
3*x^2*y + exp(y) |
x^3 + x*exp(y) |
3*x^2 + exp(y) |
3*x^2 + exp(y) |
Exact |
y |
x + y |
1 |
1 |
Exact |
x + y |
x - y |
1 |
1 |
Exact |
x*y |
x + y |
x |
1 |
Not exact generally |
Exact Differential Equations Guide
What the calculator checks
An exact differential equation links two functions, M and N. The equation is written as M dx plus N dy equals zero. It is exact when both parts come from one hidden potential function. That function is usually called F. Its x derivative gives M. Its y derivative gives N. This calculator checks that relationship with numerical derivatives.
Why exactness matters
Exactness makes a first order equation easier to solve. Instead of using a long substitution, you rebuild the potential function. Then the answer is an implicit curve. The curve is written as F of x and y equals a constant. Many physics and engineering models use this idea. It also appears in thermodynamics, vector fields, and conservative force problems.
How the result is estimated
The tool compares the partial derivative of M with respect to y against the partial derivative of N with respect to x. It does this at the selected point and over a sampled grid. A small residual means the equation behaves like an exact equation. A large residual means the two derivatives do not match well. You can tighten the tolerance for cleaner functions. You can loosen it when expressions are rough or very large.
Potential function view
The calculator estimates the potential by integrating along two simple paths. One path moves in the x direction first. Then it moves in the y direction. The second path reverses that order. For an exact equation, both paths should produce nearly the same value. The difference is shown as the path gap. A low path gap supports path independence.
Best input practice
Write multiplication clearly with an asterisk. Use x and y as variables. Use functions such as sin, cos, exp, log, sqrt, and pow. Start with a moderate grid size. Increase integration steps when you need more stable potential values. Keep graph ranges small for complex expressions. This improves speed and reduces domain errors.
FAQs
1. What is an exact differential equation?
It is a first order equation where M(x,y) dx + N(x,y) dy comes from one potential function F(x,y). The solution is written as F(x,y) = C.
2. How does the calculator test exactness?
It compares the numerical values of ∂M/∂y and ∂N/∂x. If the residual stays within the selected tolerance, the equation is treated as likely exact.
3. Does it find a symbolic potential function?
No. It estimates the potential numerically with line integrals. This works well for checking behavior, graphing levels, and building study reports.
4. Why do two paths appear in the result?
Exact equations are path independent. The two path estimates should match closely. A large path gap suggests the equation may not be exact.
5. Which functions can I enter?
You can use x, y, pi, e, powers, and common functions. Supported examples include sin, cos, tan, exp, log, sqrt, abs, and pow.
6. Why did I get a domain error?
A domain error can happen with log, sqrt, division, or trigonometric values. Adjust the point, graph range, or expression to avoid invalid values.
7. What tolerance should I choose?
Use a small tolerance, such as 0.0001, for smooth equations. Use a larger tolerance when numbers are very large or calculations are noisy.
8. Can I export my results?
Yes. Use the CSV button for spreadsheet data. Use the PDF button for a short formatted report containing the main calculated values.