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Wronskian Calculator


The calculator will use the given steps to find a Wronskian with many functions. Support up to 5 functions, namely 2x2, 3x3, etc.

Enter functions (separated by comma):


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An online 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. Read on to understand how to find wronskian using its formula and example.

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 use our wronskian 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:


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


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 calculator can determine this solution accurately by substituting the same values.

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 Wronskian Calculator Works?

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


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


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

Note: The Wronskian calculator will use the given steps to find a wronskian with several functions. Support up to 5 functions such as 2 x 2, 3 x 3.


How do you use wronskian to prove linear independence?

Let function g and f be differentiable on [x,y]. If Wronskian W (g, f) (t_0) is non-zero on [x, y] then f and g are linearly independent on [x, y]. If f and g are dependent, then the Wronskian equation is equal to zero (0) for all in [x, y].

How we can know if rows are linearly independent?

If the only linear combination of rows is equal to the zero (0) row (there is a non-linear combination of rows equal to the zero rows), then the system of these rows is called linearly independent.

What happens if the wronskian is zero?

If f and g are both solve the equation a + bx + cx = 0 for some a and b, and if Wronskian equation is equal to zero at any point in a domain, then it is zero, and functions g and f are linearly dependent. So, any two analytic functions whose wronskian is equal to zero are dependent.

What are linearly independent rows?

Linearly independent means that every column or row cannot be represented by the other columns/ rows. Hence it is independent in the whole matrix.


Use this wronskian calculator for determining the determinant and derivation of given sets, which are important for finding the wronskian of sets. Undoubtedly, calculating the wronskian manually such a difficult way. So, this wronskian solver can do all processes quickly for you without any cost.


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.

From the source of Paul Online Notes: More on the Wronskian, linearly dependent or linearly independent, Comparing Functions.