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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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