Example Data Table
| Case | M(x,y) | N(x,y) | Exact | Implicit Solution |
|---|---|---|---|---|
| 1 | 2xy + y^2 | x^2 + 2xy | Yes | x^2y + xy^2 = C |
| 2 | 3x^2 + 2y | 3x + 4y | No | Not exact |
| 3 | xy + 4x | x^2/2 + 3 | Yes | x^2y/2 + 2x^2 + 3y = C |
Formula Used
Write the equation as M(x,y)dx + N(x,y)dy = 0.
Test exactness with ∂M/∂y = ∂N/∂x.
If exact, integrate M(x,y) with respect to x.
That gives F(x,y) plus a possible y-only term.
Differentiate F(x,y) with respect to y.
Compare it with N(x,y) to find the missing correction.
Integrate that correction with respect to y.
Then write the final implicit solution as F(x,y) = C.
How to Use This Calculator
Enter each polynomial term of M(x,y) in the first group.
Enter each polynomial term of N(x,y) in the second group.
Use one row for one term.
For 2xy, enter coefficient 2, x exponent 1, and y exponent 1.
Leave unused rows blank.
Optionally add x and y values for point evaluation.
Press Solve Equation.
The result appears above the form under the header area.
Use the download buttons to save the final output.
Understanding This Exact Differential Equation Solver
An exact differential equation solver calculator helps students and teachers test whether a first order differential equation is exact. It also builds the implicit solution when the condition is satisfied. This saves time during homework, revision, and exam practice. You can enter polynomial terms for M(x,y) and N(x,y). The calculator then compares partial derivatives and forms the potential function step by step.
Why Exactness Matters
A differential equation written as M(x,y)dx + N(x,y)dy = 0 is exact when partial M over partial y equals partial N over partial x. That equality means both parts come from one potential function. Once that function is found, the solution is written as F(x,y) = C. This method is important in calculus, engineering mathematics, and mathematical modeling. It turns a difficult looking equation into a structured solving process.
How This Tool Supports Learning
This calculator is useful because it shows the entered terms, derivative comparison, exactness status, and final implicit solution. It also supports optional point evaluation for quick checking. Students can confirm class examples. Tutors can build practice sheets. Independent learners can review exact differential equation steps without searching through long notes. The export buttons are useful for saving results, sharing work, and creating printable study material.
Best Use Cases
Use this page for polynomial based exact equations, classroom demonstrations, and worked examples. It fits courses on differential equations, applied mathematics, and introductory modeling. It is also helpful for checking hand calculations before submitting assignments. When the equation is not exact, the calculator still reports the derivative mismatch clearly. That feedback helps you see where the structure fails and whether another method may be needed.
Study Smarter With Clear Structure
A good exact differential equation solver should be simple, readable, and accurate. This page keeps the layout clean and the workflow direct. Enter terms, solve, review formulas, and export the result. With repeated practice, you can recognize exact equations faster and write implicit solutions with more confidence. It is especially helpful when you want fast verification before an exam. Instead of repeating derivative algebra manually, you can focus on interpretation, accuracy, and the logic behind each exact solution step daily.
FAQs
1. What does this calculator solve?
It checks whether a first order differential equation in the form Mdx + Ndy = 0 is exact. If the condition holds, it builds the potential function and writes the implicit solution.
2. What kind of inputs does it support?
This version is designed for polynomial terms in x and y. You enter each term with a coefficient, an x exponent, and a y exponent.
3. How is exactness tested?
The calculator differentiates M with respect to y and N with respect to x. If those two derivative expressions match, the entered equation is exact.
4. What happens when the equation is exact?
The tool integrates M with respect to x, finds any missing y-only correction from N, and returns the potential function F(x,y). The final answer is shown as F(x,y) = C.
5. What happens when the equation is not exact?
The calculator reports that the equation is not exact and displays the two partial derivatives. That helps you see the mismatch before trying a different solving method.
6. Why is point evaluation included?
Point evaluation lets you check the numerical value of M, N, and the potential function at a chosen point. It is useful for verification and practice.
7. Can I export the result?
Yes. You can download a CSV summary of the result or save a PDF version. This is useful for assignments, notes, and revision sheets.
8. Is this page good for learning exact equations?
Yes. It is useful for practice because it combines the exactness test, the potential function method, example data, formulas, and a clear result area in one page.