{
  "id": "wronskian",
  "version": "30ab5c2616a2",
  "status": "published",
  "name": "Wronskian Calculator",
  "question": "What is the Wronskian?",
  "summary": "Finds the Wronskian determinant of 2 to 4 functions of one variable and whether it shows linear independence, checked numerically.",
  "category": "math",
  "subcategory": "calculus",
  "url": "https://www.acalculator.org/math/wronskian-calculator",
  "markdown": "https://www.acalculator.org/math/wronskian-calculator.md",
  "kind": "function",
  "method": "W is the determinant whose rows are the functions and their 1st to (n − 1)th derivatives, each derivative checked against difference quotients.",
  "assumptions": [
    "Radians. An answer that fails its check is not shown."
  ],
  "inputs": {
    "$schema": "https://json-schema.org/draft/2020-12/schema",
    "type": "object",
    "properties": {
      "f": {
        "title": "Functions",
        "description": "2 to 4 functions of one variable, separated by commas.",
        "type": "string",
        "maxLength": 240
      }
    }
  },
  "outputs": {
    "w": {
      "label": "Wronskian W",
      "description": "The determinant of the functions and their derivatives.",
      "format": "math"
    },
    "verdict": {
      "label": "Linear independence",
      "description": "What W says about linear independence.",
      "format": "text"
    }
  },
  "defaultAnswer": {
    "inputs": {
      "f": "e^x, e^(-x)"
    },
    "outputs": {
      "w": "-2",
      "verdict": "Linearly independent: W is not 0 everywhere."
    },
    "text": "The Wronskian of e^x, e^(-x) is -2."
  },
  "examples": [
    {
      "given": {
        "f": "e^x, e^(-x)"
      },
      "expect": {
        "w": "-2"
      },
      "source": "Trench (2013), Elementary Differential Equations, 5.1, Example 5.1.5 (a)"
    },
    {
      "given": {
        "f": "x^2, 1/x^2"
      },
      "expect": {
        "w": "-4/x"
      },
      "source": "Trench (2013), Elementary Differential Equations, 5.1, Example 5.1.5 (c)"
    }
  ],
  "sources": [
    "William F. Trench, Elementary Differential Equations with Boundary Value Problems, section 5.1 Homogeneous Linear Equations (LibreTexts): https://math.libretexts.org/Bookshelves/Differential_Equations/Elementary_Differential_Equations_with_Boundary_Value_Problems_(Trench)/05%3A_Linear_Second_Order_Equations/5.01%3A_Homogeneous_Linear_Equations",
    "OpenStax, Calculus Volume 3, section 7.1 Second-Order Linear Equations: https://openstax.org/books/calculus-volume-3/pages/7-1-second-order-linear-equations"
  ],
  "related": [
    "derivative",
    "determinant",
    "partial-derivative"
  ],
  "changelog": []
}