Enter Equation Details
Example Data Table
| Equation Type | Inputs | Initial Values | Expected Form |
|---|---|---|---|
| First Order Linear | p = 3, q = 6 | x₀ = 0, y₀ = 2 | y = q / p + Ce-px |
| Exponential | k = 2 | x₀ = 0, y₀ = 5 | y = Cekx |
| Logistic | r = 0.8, K = 100 | x₀ = 0, y₀ = 10 | y = K / (1 + Ae-rx) |
| Second Order | a = 1, b = 5, c = 6 | y₀ = 2, y'₀ = 0 | Based on characteristic roots |
Formula Used
The calculator chooses formulas according to the selected equation family.
- y' = ky gives y = Ce^(kx).
- y' + py = q gives y = q/p + Ce^(-px), when p is not zero.
- y' = kyⁿ gives y = [(1-n)(kx+C)]^(1/(1-n)), when n is not one.
- y' = ry(1-y/K) gives y = K/(1+Ae^(-rx)).
- ay'' + by' + cy = 0 uses ar² + br + c = 0.
- ax²y'' + bxy' + cy = 0 uses am(m-1)+bm+c=0.
How to Use This Calculator
- Select the equation family that matches your problem.
- Enter the needed coefficients for that family.
- Add initial values to create a particular solution.
- Enter the x value where you want evaluation.
- Press the calculate button.
- Review the general solution, particular solution, steps, and check line.
- Use the CSV or PDF button to save the result.
Differential Equation Solving Guide
Purpose
Differential equations describe changing systems. They connect a function with one or more derivatives. This calculator helps you solve common forms quickly. It gives a general solution, a particular solution, and an evaluation point.
Model Selection
Start by matching your equation to a supported family. Use exponential mode for direct growth or decay. Use linear mode when the derivative plus a multiple of the function equals a constant. Use logistic mode for limited growth. Use second order mode for vibration, damping, and repeated root cases.
Initial Conditions
Initial conditions change a broad solution into one exact curve. First order models need one value, usually y at x zero. Second order models need two values. They often use y and y prime at the same starting point.
Constants
The symbol C represents an unknown integration constant. After you enter initial values, the calculator finds this constant. In second order models, two constants may appear. They represent the two independent parts of the solution.
Checking Work
A useful solution should satisfy the original equation. The calculator shows a residual or characteristic check. A residual near zero means the equation is consistent. A discriminant explains which second order form is used.
Practical Use
Students can compare homework answers. Teachers can prepare examples. Engineers can test simple dynamic models. Analysts can explore growth, decay, and limiting behavior. The exported files make reports easier to save and share.
Limits
No simple tool solves every possible differential equation. Some problems need numerical solvers, series methods, Laplace transforms, or symbolic algebra. This page focuses on common closed form families. It is best for checking structure, constants, and basic evaluation.
Frequently Asked Questions
Can this solve every differential equation?
No. It solves several common closed form families. More complex equations may need numerical methods, transforms, or a symbolic algebra system.
What is a general solution?
A general solution contains arbitrary constants. It represents a family of possible curves that satisfy the differential equation.
What is a particular solution?
A particular solution uses initial conditions to calculate constants. It gives one exact curve from the broader general family.
Why does second order mode need y prime?
A second order equation usually has two constants. It needs two initial conditions, such as position and velocity, to find both constants.
What does the discriminant mean?
The discriminant classifies roots of the characteristic equation. Positive gives two real roots, zero gives repeated roots, and negative gives complex roots.
Why can some separable answers be unavailable?
Some power solutions leave the real-valued domain at the selected evaluation point. Try another point or review the equation domain.
Can I export the result?
Yes. After calculation, use the CSV button for spreadsheet data or the PDF button for a printable report.
Is this useful for conversion pages?
Yes. It can support educational conversion workflows where equations, constants, and evaluated outputs must be compared clearly.