Calculator Input
Enter the equation details. The calculator checks ODE/PDE family, order, degree, linearity, homogeneity, and second-order PDE type.
Formula Used
aₙ(x)y⁽ⁿ⁾ + aₙ₋₁(x)y⁽ⁿ⁻¹⁾ + ... + a₁(x)y' + a₀(x)y = g(x)
aₙ(x)y⁽ⁿ⁾ + ... + a₀(x)y = 0
Auₓₓ + Buₓᵧ + Cuᵧᵧ + ... = 0, use
D = B² − 4AC.
If D < 0, it is elliptic.
If D = 0, it is parabolic.
If D > 0, it is hyperbolic.
How to Use This Calculator
- Enter the differential equation in readable notation.
- Select whether it is an ordinary or partial differential equation.
- Enter the dependent and independent variables.
- Choose the highest order and degree.
- Mark nonlinear checks, such as derivative products or powers.
- For a second-order PDE, enter coefficients A, B, and C.
- Click the classify button to view the result above the form.
- Use the CSV or PDF button to export your results.
Example Data Table
| Equation | Family | Order | Degree | Linearity | Extra Class |
|---|---|---|---|---|---|
y'' + 3y' + 2y = 0 |
ODE | 2 | 1 | Linear | Homogeneous |
y' = xy |
ODE | 1 | 1 | Linear | Separable |
y'' + y² = x |
ODE | 2 | 1 | Nonlinear | Nonhomogeneous |
uₓₓ + uᵧᵧ = 0 |
PDE | 2 | 1 | Linear | Elliptic |
uₜₜ − c²uₓₓ = 0 |
PDE | 2 | 1 | Linear | Hyperbolic |
Understanding Differential Equation Classification
Why Classification Matters
A differential equation describes change. It links an unknown function with its derivatives. Classification tells you what methods may work before solving. It also prevents wasted effort on a wrong technique.
Core Equation Family
The first check is the equation family. An ordinary differential equation uses one independent variable. A partial differential equation uses two or more independent variables. This difference changes the notation, data needs, and possible solution methods.
Order, Degree, and Linearity
Order is the highest derivative present. Degree is the power of the highest derivative, when the equation is polynomial in derivatives. If radicals, fractions, or transcendental functions contain derivatives, the degree is normally not defined. This calculator reports that case clearly.
Linearity is another important test. A linear equation has the dependent variable and its derivatives only to the first power. They are not multiplied together. Their coefficients may depend on independent variables. If the dependent variable appears inside sine, exponential, logarithmic, or a product with itself, the equation is nonlinear.
Homogeneity and PDE Type
Homogeneity depends on the right side. For a linear equation, zero forcing gives a homogeneous form. A nonzero source term gives a nonhomogeneous form. This matters because complementary and particular solutions are handled differently.
For second order equations in two variables, the PDE type can be elliptic, parabolic, or hyperbolic. The test uses the discriminant B squared minus four A C. Negative values often model steady states. Zero values often model diffusion. Positive values often model wave behavior.
Practical Study Use
The calculator also records separable, exact, autonomous, initial value, and boundary value clues. These labels help organize a solution plan. They do not replace algebraic verification. They give a structured review of the equation.
Use the chart to compare key indicators. Export the result for notes, worksheets, or reports. Always rewrite the equation in standard form when possible. Then check each classification again. Clear classification makes advanced solving faster, safer, and easier to explain.
Include units and variable names when teaching or documenting the work. Small notation changes can alter the class. For example, y times y prime is nonlinear, but x times y prime can still be linear. Such details matter in exams, modeling tasks, and numerical planning. Review assumptions before choosing software.
FAQs
1. What does this calculator classify?
It classifies differential equations by family, order, degree, linearity, homogeneity, autonomous status, PDE type, and problem data form.
2. Can it solve the differential equation?
No. It focuses on classification. Use the result to choose a suitable solving method, numerical method, or study path.
3. When is degree not defined?
Degree is not defined when derivatives appear inside radicals, fractions, logarithms, trigonometric terms, or other non-polynomial expressions.
4. What makes an equation nonlinear?
An equation is nonlinear when the dependent variable or its derivatives are powered, multiplied together, or placed inside nonlinear functions.
5. What is a homogeneous differential equation?
For linear equations, homogeneous usually means the right side or forcing term is zero after writing the equation in standard form.
6. How is a second-order PDE classified?
Use D = B² − 4AC. Negative means elliptic. Zero means parabolic. Positive means hyperbolic for the selected standard form.
7. Why does notation matter?
Small notation changes can change the class. A coefficient depending on x may be linear, while one depending on y may be nonlinear.
8. Can I export the result?
Yes. Use the CSV button for spreadsheet data. Use the PDF button for a printable classification summary.