What does a zero Wronskian mean?

If the Wronskian is identically zero everywhere on an interval it suggests the functions may be linearly dependent, though the converse is not always true. A Wronskian that is non-zero at even one point guarantees the functions are linearly independent.

Which functions can I enter?

You can enter polynomials (x^2, 3x+1), exponentials (e^x, e^(2x)), trigonometric functions (sin(x), cos(2x), tan(x)), natural logarithm (ln(x)), and basic combinations of these using +, -, *, /.

How is the 3-function Wronskian computed?

For three functions f, g, h the Wronskian is the 3×3 determinant with rows [f, g, h], [f', g', h'], [f'', g'', h'']. The tool computes first and second derivatives of each function, then expands the determinant via cofactor expansion along the first row.

Is this tool exact or numerical?

Derivatives are computed symbolically using pattern-matching rules for the supported function types. The determinant is then simplified and displayed symbolically. For complex expressions the displayed form may not be fully reduced.

Can I use constants in my functions?

Yes, you can use numeric coefficients such as 3*sin(2*x) or 5*e^(3*x). Multiply with * explicitly — 3sin(x) without the * is not supported.

Wronskian Calculator

Name: Wronskian Calculator
Availability: InStock
Author: Nham Vu

Enter 2 or 3 functions to compute the Wronskian determinant and test for linear independence, with full step-by-step working.

Functions

Number of Functions

Variable

f₁(x)

f₂(x)

Examples

Enter your functions and click Compute Wronskian

Summary

Enter 2 or 3 functions to compute the Wronskian determinant and test for linear independence, with full step-by-step working.

How it works

Enter 2 or 3 functions in the input fields (e.g. x^2, sin(x), e^x).
The calculator computes the first derivative (and second derivative for 3 functions) of each function.
It assembles the Wronskian matrix and expands the determinant symbolically.
The result is displayed along with each intermediate step so you can follow the working.
If the Wronskian evaluates to zero the functions may be linearly dependent; a non-zero result confirms independence.

Use cases

Verify that a pair of solutions to a second-order ODE form a fundamental set.
Check whether three candidate solution functions are linearly independent before using variation of parameters.
Practice computing 2×2 or 3×3 determinants of derivative matrices in a differential equations course.
Confirm that a basis for a solution space spans the full space by testing its Wronskian.
Debug hand calculations by comparing step-by-step output against your own working.

Frequently Asked Questions

Last updated: 2026-07-26 · Reviewed by Nham Vu