## Introduction to Wronskian Calculator:

Wronskian calculator is the best online source that helps you in solving the linear differential equation. It uses the differentiate method inside the determinant of the function to get solution in less than a minute.

It is a beneficial tool for students, teachers or researchers as it gives the solution of ODEs equation even for complex functions without taking any external assistance.

## What is Wronskian?

Wronskian method is a process in which you determine whether the **linear differential function** or differential equation is linearly dependent or independent on linear algebra. The wroniskan method is represented with the symbol “W''.

For f1, f2,...fn functions, it uses the differential and determinate matrix method in which if the wronskian method solution is non zero then your function is linearly independent. On the other hand, if the given function solution is zero then the function is linearly dependent.

## Formula of Wronskian:

The **wronskian method formula** is based on the function differentiation f1(x), f2(x),…, fn(x) and the determinate matrix where you put the function and solve it.

$$ W(f_1, f_2,..., f_n)(x) \;=\; \left(\begin{matrix} f_1(x) & f_2(x) & \cdots & f_n(x) \\ f_1’(x) & f_2’(x) & … & f_n’(x) \\ \vdots & \vdots & \ddots & \vdots \\ f_1^{(n-1)}(x) & f_2^{(n-1)}(x) & \cdots & f_n^{(n-1)}(x) \end{matrix} \right) $$

## How to Calculate the Wronskian?

For the **calculation of Wronskian** of a set of linear functions, the wronskian determinant calculator uses the determinant method to find whether a set of functions is linearly independent. Here's a stepwise guide on how to calculate the Wronskian manually:

**Step 1:**

Identify the linear functions f1(x), f2(x),…,fn(x) and variable of differentiation.

**Step 2:**

Find the first derivative of each function fi(x), calculate fi′(x), fi′′(x),…,fi(n − 1)(x).

**Step 3:**

According to the numbers of functions, add the differential function value f`(xi) and the linear differential function f(xi) of determinate matrix. For example, 2 by 2 determinant matrix for wroiskan is:

$$ W (f_1, f_2)(x) \;=\; \biggr|\begin{matrix} f_1 (x) & f_2(x) \\ f_1’(x) & f_2’(x) \\ \end{matrix} \biggr| $$

**Step 4:**

Solve the determinant whether the given matrix is 2 by 2 determinate or 2 by 3 determinant matrix.

$$ W (f_1, f_2)(x) \;=\; \biggr|\begin{matrix} f_1(x) & f_2(x) \\ f_1’(x) & f_2’(x) \\ \end{matrix} \biggr| \;=\; f_1(x) . f_2’(x) - f_2(x) . f_1’(x) $$

**Step 5:**

After simplification, you get the solution of the Wronskian method problem which determines whether the linear function is independent or dependent.

## Solved Example of Wronskian Method:

A solved **example** of the Wronskian method is given below to understand how the Wronskian calculator with steps works.

### Example: Find the wronskian of the following:

$$ f_1 \;=\; x^2 + 4,\; f_2 \;=\; sin(2x) $$

**Solution**:

The given function is,

$$ f_1 \;=\; x^2 + 4,\; f_2 \;=\; sin(2x) $$

Differentiate the function f1(x) and f2(x) with respect to x.

$$ \frac{d}{dx} (x^2 + 4) \;=\; 2x $$

$$ \frac{d}{dx} (sin(2x)) \;=\; 2\; cos (2x) $$

As the given function has 2 by 2 matrix so the require determinant matrix become,

$$ W (f_1, f_2)(x) \;=\; \biggr|\begin{matrix} f_1(x) & f_2(x) \\ f_1’(x) & f_2’(x) \\ \end{matrix} \biggr| $$

Now make a determinate matrix and put the linear function and derivative function value,

$$ W (f_1, f_2)(x) \;=\; \biggr|\begin{matrix} x^2 + 4 & sin(2x) \\ 2x & 2\;cos(2x) \\ \end{matrix} \biggr| $$

Solve the determinant matrix as per the rule of matrix determination,

$$ W (f_1, f_2)(x) \;=\; \biggr|\begin{matrix} f_1(x) & f_2(x) \\ f_1’(x) & f_2’(x) \\ \end{matrix} \biggr| \;=\; f_1(x) . f_2’(x) - f_2(x) . f_1’(x) $$

$$ W (f_1, f_2)(x) \;=\; \biggr|\begin{matrix} x^2 + 4 & sin(2x) \\ 2x & 2\;cos(2x) \\ \end{matrix} \biggr| \;=\; 2x^2 cos (2x) - 2x\; sin(2x) + 8\; cos(2x) $$

The result of given linear differential function using wronkisan method is,

$$ W (f_1, f_2)(x) \;=\; 2x^2\; cos(2x) - 2x\; sin(2x) + 8\; cos(2x) $$

## How to Use the Wronskian Calculator 3x3?

The Wronskian matrix calculator has a simple design that helps to solve the given linearly independent function. You just need to put the input value.

**Enter the linear differential**function in the input field of wronskian method calculator.- Choose the variable of differentiation from the given list of wronskian determinant calculator.
- Check your given input function to get the correct solution of the linear differential function question.
- Click on the "Calculate" button to get the result of the given linear differential function problems.
- If you want to understand the calculation process of our Wronskian method calculator then use the load example option and get its solution.
- Click the “Recalculate” button for the calculation of more examples of linear differential functions with the solution.

## Final Result of Wronskian Calculator:

Wronskian linear independence calculator provides you with a **solution** as per your input value problem. It contain as:

**Result Box**:

Click on the result button so that you get the solution of your linear differential function.

**Steps Box**:

When you click on the steps option, you get the step by step result of linear differential questions.

## Benefits of Wronskian Matrix Calculator:

The Wronskian differential equations calculator has many benefits that you get when you use it to solve Wronskian method differential equation problems. Our tool only takes the input function and provides you a solution instantly. These benefits are

- It is a
**reliable tool**as it gives you accurate solutions of linear differential equation problems. - It is a speedy tool that provides solutions to the given wronskian method problems in a few seconds.
- It is a learning tool as it helps you to learn the wronskian method for linear differential function online.
- It is a handy tool that can solve various types of linear differential equation problems easily.
- Wronskian linear independence calculator is a free tool that allows you to calculate the linear differentiation functions many times without spending anything.
- Wronskian determinant calculator is an easy-to-use tool, anyone or even a beginner can easily use it to get the solution of wronskian method problems.