Wronskian Calculator

Wronskian calculator will allow you to determine the wronskian of the given set of functions. The calculator also takes the determinant then calculates the derivative of all functions.

What is the Wronskian?

In mathematics, the Wronskian is a determinant introduced by Józef in the year 1812 and named by Thomas Muir. It is used for the study of differential equations wronskian, where it shows linear independence in a set of solutions.

In other words, the Wronskian of the differentiable functions g and f is W (f, g) = fg’ – f’g. For complex or real valued functions f_1, f_2, f_3, . . . , f_n, which are n – 1 times differentiable on the interval L, so the wronskian formula W(f_1, f_2, f_3, . . . , f_{n-1},  f_n) as a function on L is defined by

W (f_1, f_2, …, f_n) (x) =

\( \begin{vmatrix} f_1(x) & f_2(x)& … & f_n(x) \\ f’_1(x) & f’_2(x) & … & f’_n(x) \\ . & . & . & . \\ f_1^{n-1} (x) & f_2^{n-1} (x) & … & f_n^{(n-1)} (x) \end{vmatrix} \)

How to Calculate Wronskian?

You can this calculator for taking the determinant and derivative of the given set for finding the wronskian. If you want to do all calculations manually then see the example below:

Example:

To find the Wronskian of: (x^2+4),sin (2x),cos (x)

Solution:

The given set of functions is:

\({f_1 = (x^2+4), f_2 = sin (2x), f_3 = cos (x)}\)

Then, the Wronskian formula is given by the following determinant:

W (f_1, f_2, f_3) (x) =

\begin{vmatrix} f_1(x) & f_2(x) & f_3(x) \\ f’_1(x) & f’_2(x) & f’_3(x) \\ f’’_1 (x) & f’’_2 (x) & f’’_3 (x) \end{vmatrix}

In our case:

W (f_1, f_2, f_3) (x) =

\begin{vmatrix} (x^2+4) & (sin (2x)) & cos (x) \\ (x^2+4)’ & (sin (2x))’ & cos (x)’ \\ (x^2+4)’’ & (sin (2x))’’ & cos (x)’’ \end{vmatrix}

Now, find the derivative

W (f_1, f_2, f_3) (x) =

\begin{vmatrix} (x^2+4) & (sin (2x)) & cos (x) \\ 2x & (2cos (2x)) & -sin (x) \\ 2 & (-4 sin (2x)) & – cos (x) \end{vmatrix}

Then find the determinant:

W (f_1, f_2, f_3) (x) =

\begin{vmatrix} (x^2+4) & (sin (2x)) & cos (x)\\ 2x & (2cos (2x)) & -sin (x)\\2 & (-4 sin (2x)) & – cos (x) \end{vmatrix}

= 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 function f_i is linearly dependent, then the columns of Wronskian will also be dependent because differentiation is a linear operation, so Wronskian disappears.

Thus, it can be used to illustrate that a set of differentiable functions is independent of the interval that does not vanish identically.

How This Calculator Works?

The wronskian solver can find the wronskian by the determinant of given functions by following these instructions:

Input:

• Enter functions with respect to any variable from the drop-down list.
• Click on the calculate button for wronskian calculations.

Output:

• The calculator displays all wronskian functions.
• It provides the Wronskian by the derivation of given functions with stepwise calculations.

Reference:

From the source of Wikipedia: The Wronskian and linear independence, Application to linear differential equations, Generalized Wronskians.

From the source of ITCC Online: Linear Independence and the Wronskian, Abel’s Theorem, Linearly dependent.