Wronskian Calculator

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Wronskian calculator allows 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 for Wronskian 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.