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Wronskian Calculator

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Wronskian Calculator

The Wronskian calculator allows you to determine the Wronskian of a set of functions. The calculator computes derivatives of the functions, forms the determinant, and calculates the Wronskian automatically.

What is the Wronskian?

In mathematics, the Wronskian is a determinant introduced by Józef Hoene-Wronski in 1812 and later named by Thomas Muir. It is commonly used in the study of differential equations to test whether a set of solutions is linearly independent.

For two differentiable functions f and g, the Wronskian is defined as:

W(f, g) = f·g' - f'·g

For a set of n functions f₁, f₂, …, fₙ, all (n-1) times differentiable on an interval L, the Wronskian is given by the determinant:

\[ W(f_1, f_2, \dots, f_n)(x) = \begin{vmatrix} f_1(x) & f_2(x) & \dots & f_n(x) \\ f_1'(x) & f_2'(x) & \dots & f_n'(x) \\ \vdots & \vdots & \ddots & \vdots \\ f_1^{(n-1)}(x) & f_2^{(n-1)}(x) & \dots & f_n^{(n-1)}(x) \end{vmatrix} \]

How to Calculate the Wronskian

You can either use this calculator or compute it manually using derivatives and determinants. Here's an example:

Example:

Find the Wronskian of the functions:

f₁(x) = x² + 4
f₂(x) = sin(2x)
f₃(x) = cos(x)

Solution:

Step 1: Form the Wronskian determinant:

\[ W(f_1, f_2, f_3)(x) = \begin{vmatrix} f_1 & f_2 & f_3 \\ f_1' & f_2' & f_3' \\ f_1'' & f_2'' & f_3'' \end{vmatrix} = \begin{vmatrix} x^2+4 & \sin(2x) & \cos(x) \\ 2x & 2\cos(2x) & -\sin(x) \\ 2 & -4\sin(2x) & -\cos(x) \end{vmatrix} \]

Step 2: Compute the determinant to get the Wronskian:

\[ W(f_1, f_2, f_3)(x) = 4x^2 \cos^3(x) - 6x^2 \cos(x) + 12x \sin^3(x) - 12x \sin(x) + 12 \cos^3(x) - 24 \cos(x) \]

The Wronskian and Linear Independence

If the functions are linearly dependent, the columns of the Wronskian determinant are also dependent, and the Wronskian will vanish. A non-zero Wronskian on an interval indicates that the set of functions is linearly independent.

How This Calculator Works

Input:

Enter the set of functions with respect to a variable (e.g., x) using the input fields or drop-down menu.
Click "Calculate" to find the Wronskian.

Output:

Step-by-step derivatives of each function.
The determinant showing the Wronskian of the given functions.
The final Wronskian value as a simplified expression.

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