Analyze M and N carefully. Verify exactness and assemble solutions. Built for guided equation solving with stepwise clarity.
| M(x,y) | N(x,y) | ∂M/∂y | ∂N/∂x | ∫M dx | Correction term | Sample point |
|---|---|---|---|---|---|---|
| 2*x*y + 3 | x^2 + 4*y | 2*x | 2*x | x^2*y + 3*x | 2*y^2 | (2, 1) |
| y*cos(x) + 2*x | sin(x) + 5*y^2 | cos(x) | cos(x) | y*sin(x) + x^2 | (5/3)*y^3 | (1, 2) |
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 F(x,y) exists such that:
∂F/∂x = M(x,y)
∂F/∂y = N(x,y)
First integrate M with respect to x:
F(x,y) = ∫M(x,y)dx + g(y)
Next differentiate that result with respect to y.
Compare it with N(x,y) to find the missing term g(y).
The final implicit solution is:
F(x,y) = C
Enter the M(x,y) term and the N(x,y) term from your equation.
Provide the partial derivative of M with respect to y.
Provide the partial derivative of N with respect to x.
Add a sample x and y value for a quick exactness check.
Enter the integral of M with respect to x.
Then enter the correction term that depends only on y.
Click the solve button to view the exactness test.
The tool also assembles the potential function and implicit solution.
Use the CSV button to export the result table.
Use the PDF button to print or save the result as a PDF file.
Exact differential equations appear often in calculus, physics, and engineering. This calculator helps you organize the solving process clearly. It checks whether the mixed partial conditions match at a sample point. That quick test gives a useful confirmation before you continue.
The tool is designed for guided solving. It does not hide the mathematics. Instead, it lets you enter each important part of the workflow. You provide M(x,y), N(x,y), and their partial derivatives. Then you enter the integrated M term and the correction term in y. This approach keeps every step visible.
Students can use the calculator to verify homework steps. Teachers can use it to demonstrate the method in class. Self learners can use it to compare manual work with a structured result. Because the output is shown directly above the form, you can review the answer quickly and revise any field without losing the page flow.
The example table adds a practical starting point. It shows how exact equations are assembled and how the potential function is formed. The export options are useful when you want to save attempts, share worked examples, or print a clean solution summary for study notes.
This page also includes formula guidance and concise usage steps. That makes it useful not only as a calculator, but also as a learning companion. When the exactness condition holds, the solver helps combine the integrated expression and correction term into the final implicit solution. This keeps the method organized, repeatable, and easier to understand.
It is a first order equation written as M(x,y)dx + N(x,y)dy = 0, where a potential function F(x,y) exists and satisfies both partial derivative conditions.
Compute ∂M/∂y and ∂N/∂x. If they are equal on the domain of interest, the equation is exact and can be solved using a potential function.
The tool uses your derivative entries to verify the exactness condition. This keeps the method transparent and supports manual learning.
General symbolic integration is broad and case dependent. This guided format lets you control the exact integral expression used in your solution.
After integrating M with respect to x, a missing term depending only on y may remain. That extra part is the correction term.
No. The sample point offers a quick numerical check. A full proof still depends on the symbolic equality of ∂M/∂y and ∂N/∂x.
Yes. You can export the visible result as a CSV file or use the PDF button to print or save the result section.
It is useful for students, tutors, and professionals reviewing exact equation steps. It supports practice, teaching, and documentation.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.