Calculator inputs
Use coefficient-based forms for fast classroom checks, tutorials, and worked examples.
Example data table
These examples show how the tool can be used across the three supported modes.
| Mode | Input form | Sample coefficients | Expected integrating factor |
|---|---|---|---|
| Linear ODE | y′ + P(x)y = Q(x) | P(x)=x, Q(x)=2, y(0)=1 | μ(x)=ex²/2 |
| Special μ(x) | (My − Nx)/N = f(x) | f(x)=2x | μ(x)=ex² |
| Special μ(y) | (Nx − My)/M = g(y) | g(y)=y+1 | μ(y)=ey²/2 + y |
Formula used
Linear form: For y′ + P(x)y = Q(x), the integrating factor is μ(x) = e∫P(x)dx. Then y = μ(x)−1[C + ∫μ(x)Q(x)dx].
Special factor depending on x: If (My − Nx)/N = f(x), then μ(x) = e∫f(x)dx.
Special factor depending on y: If (Nx − My)/M = g(y), then μ(y) = e∫g(y)dy.
How to use this calculator
- Select the solving mode that matches your differential equation.
- Enter the sampling interval and number of steps.
- For linear equations, enter P(x), Q(x), and the initial value.
- For special exactness checks, enter the simplified ratio polynomial.
- Press Calculate to show results above the form.
- Review the steps, graph, and exportable value table.
FAQs
1. What is a special integrating factor?
It is a multiplier that turns a difficult first-order differential equation into an exact or directly solvable form. This tool focuses on linear equations and exactness ratios depending only on one variable.
2. When should I use the linear mode?
Use linear mode when the equation can be written as y′ + P(x)y = Q(x). The calculator builds μ(x), applies the standard solution formula, and samples the resulting curve.
3. What does the μ(x) special mode test?
It tests the classic condition (My − Nx)/N = f(x). When that ratio depends only on x, the integrating factor is e raised to the integral of that ratio.
4. What does the μ(y) special mode test?
It tests whether (Nx − My)/M reduces to a function of y only. If it does, the integrating factor depends only on y.
5. Why are coefficients used instead of full algebra input?
Coefficient input keeps the tool fast, stable, and easy to validate in classroom settings. It also supports quick examples, worksheet checks, and smooth plotting without a symbolic parser.
6. Are the linear solutions exact?
The integrating factor formula is exact, but the displayed solution curve uses numerical accumulation for the integral term. With reasonable step counts, the sampled values are usually very good for study and checking.
7. What does the graph show?
For linear mode, it plots both μ(x) and the estimated solution y(x). For the special modes, it plots the detected ratio and the integrating factor over the selected interval.
8. Can I export the table for notes?
Yes. Use the CSV button for spreadsheet work and the PDF button for neat revision sheets or assignment attachments. The export uses the same table shown on screen.