## Wronskian Calculator

Enter the values of the functions in the Wronskian calculator to find their determinant and check their linear independency.

## How to use this calculator?

There is only one step to use this calculator: enter the functions and click calculate

## What is the Wronskian determinant?

The Wronskian is named after the Polish mathematician Józef Hoene-Wroński. It is a mathematical method used to check if a set of functions is linearly dependent or independent. It involves a determinant that encompasses the functions and their derivatives.

If the Wronskian's value is zero within a certain range, the functions are linearly dependent; otherwise, they are linearly independent.

## Wronskian formula:

Given n differentiable functions f1 (x),f2 (x),…, fn (x), their Wronskian W is defined as the determinant of the n×n matrix whose ith row consists of the i−1th derivatives of the functions:

## How to find the Wronskian determinant?

Given two functions y1 (x) and y2 (x):

• Write down y1 and y2.
• Calculate the first derivatives: y1’ and y2’.
• Create a 2x2 matrix where the first row contains the functions y1 and y2, and the second row contains their corresponding first derivatives.
• Evaluate the determinant of this matrix.

Mathematically, the Wronskian W(y1,y2) is:

### Solved example:

Let’s determine the Wronskian of e^x and xe^x.

Step 1: Write the given functions.

y1 = e^x

y2 = xe^x

Step 2: Calculate the first derivatives.

y1’ = e^x

y2’ = (x+1) e^x

Step 3: Construct the matrix.

Step 4: Evalvulate the determinant.

= e^x (xe^x + e^x) - xe^x (e^x)

= e^2x - xe^2x + xe^2x

= e^2x

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