Wronskian Calculator

Summary

The Wronskian is a determinant used in the theory of differential equations to test whether a set of functions is linearly independent on an interval. For two functions f and g, the Wronskian is W = f·g' − f'·g; for three functions it extends to a 3×3 determinant. A non-zero Wronskian at any point confirms the functions are linearly independent. This tool evaluates the Wronskian symbolically using common function rules and displays every derivative and determinant expansion step.

How it works

1. Enter 2 or 3 functions in the input fields (e.g. x^2, sin(x), e^x).
2. The calculator computes the first derivative (and second derivative for 3 functions) of each function.
3. It assembles the Wronskian matrix and expands the determinant symbolically.
4. The result is displayed along with each intermediate step so you can follow the working.
5. 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