Integral Curve Input Panel
Example Data Table
| Model | A | B | C or M | x0 | y0 | Target x | Expected use |
|---|---|---|---|---|---|---|---|
| dy/dx = A*x + B*y + C | 2 | -0.4 | 1 | 0 | 3 | 2 | Linear field with forcing |
| dy/dx = K*y | 0.25 | 0 | 0 | 1 | 4 | 5 | Growth or decay curve |
| dy/dx = A*x^M + B | 3 | 2 | 2 | 0 | 1 | 3 | Power based slope field |
Formula Used
An integral curve solves a differential equation and passes through a given point. The general condition is y(x0) = y0. The calculator first builds the symbolic curve for the chosen model. Then it substitutes the initial point to find the constant.
Direct model: For dy/dx = A*x + B, the formula is y = (A/2)x² + Bx + C.
Exponential model: For dy/dx = K*y, the formula is y = Ce^(Kx).
Affine linear model: For dy/dx = A*x + B*y + C, the equation is solved as y′ - By = Ax + C.
Power model: For dy/dx = A*x^M + B, the formula is y = A*x^(M+1)/(M+1) + Bx + C, when M is not -1.
Homogeneous model: For dy/dx = A*y/x, the formula is y = C|x|^A.
How to Use This Calculator
- Select the differential model that matches your problem.
- Enter the required coefficients.
- Enter the initial point values x0 and y0.
- Add a target x value for evaluation.
- Choose table steps for the curve sample.
- Press the calculate button.
- Review the equation, slope, constant, and RK4 check.
- Download the results as CSV or PDF.
Integral Curves in Differential Equations
What an Integral Curve Means
An integral curve is a path that follows a differential equation. Each point on the curve agrees with the slope rule. The rule may depend on x, y, or both. A starting point fixes the final curve. Without that point, many curves may satisfy the same equation.
Why Initial Values Matter
The initial value gives one exact member of a family. It is written as y(x0) = y0. This condition finds the integration constant. That constant shifts or scales the curve. The result becomes a single equation, not just a general family.
Supported Model Types
This tool supports several common forms. It handles direct x based slopes. It handles exponential y based slopes. It also handles affine linear fields with x and y terms. Power rules and homogeneous y over x rules are included for wider classroom use.
Exact and Numeric Checking
The calculator shows the exact curve when the model has a closed form. It also computes a Runge Kutta check. This check follows the slope field step by step. The error column compares both outputs. Small error values support the computed equation.
Reading the Output
The equation row gives the integral curve. The constant row shows the value found from the initial point. The target result gives y at the selected x value. The slope result shows dy/dx at that same location. The sample table helps review the curve across the interval.
Learning and Practical Uses
Students can test homework answers. Teachers can create quick examples. Engineers can inspect simple rate models. Analysts can explore growth, decay, and forced change. The CSV file helps with spreadsheets. The PDF file helps with notes, reports, and printed records.
Domain Awareness
Some models have restrictions. Logarithmic forms cannot use x equal to zero. Negative bases may fail with non-integer powers. The y over x model is also undefined at x equal to zero. Always check the notes section when results look unusual.
FAQs
What is an integral curve?
It is a curve whose tangent slope follows a given differential equation at every point. An initial condition selects one curve from the full family.
What does the initial value do?
The initial value fixes the integration constant. It ensures the final curve passes through the selected point exactly.
Can this solve every differential equation?
No. It solves common closed form models. It also gives a numerical RK4 check for the supported forms.
What is the RK4 y check?
RK4 is a numerical method. It follows the slope field in steps and compares its estimate with the exact formula.
Why is a result not defined?
A model may have a domain problem. Common causes include x equal to zero, logarithms, or fractional powers of negative values.
Which model should I choose?
Match the visible structure of your equation. Choose x only, y only, affine x and y, power x, or y divided by x.
Can I download the table?
Yes. Use the CSV button for spreadsheet data. Use the PDF button for a simple report with the main equation and table.
Is the target x required?
Yes. It tells the calculator where to evaluate the integral curve, slope, and final y value.