The Wronskian is a determinant formulated by Polish mathematician and philosopher Józef Maria Hoëne-Wroński. It is used to find if two or more functions are linearly independent. Functions that are linearly dependent are multiples of each, whereas linearly independent ones are not. If the Wronskian is zero at all points, which means it vanishes everywhere, then the functions are linearly dependent. In mathematical terms, for two functions f and g, this means that W(f, g) = 0. If the Wronskian is zero only at certain points, linear dependence has not been proven. To calculate the Wronskian, you need to know how to use determinants and how to find the derivatives of functions.

## Two Functions

Use the Wronskian formula for two functions, as shown on the left. The determinant is calculated using the formula W(f, g) = fg' -- gf '. If this is equal to zero at all values, the functions f and g are multiples of each other and hence are linearly dependent.

Solve the Wronskian for two functions. As an example, for e^x and e^2x, the determinant is as shown on the left. The derivative for e^x is e^x, and the derivative for e^2x is 2e^2x. The Wronskian is e^x * 2e^2x -- e^2x * e^x.

Simplify the expression in step two. This is equal to 2e^3x -- e^3x. So W(e^x, e^2x) = e^3x. Since this is never zero for any value of x, the two functions are linearly independent.

## Three Functions

Use the Wronskian for three functions. The determinant for functions f, g, and h is W(f, g, h) = f(g'h'' -- h'g'') - g(f 'h'' -- h'f '') +h(f 'g'' -- g'f'').

Solve the Wronskian for three functions. As an example, for 1, x, and x^2, the determinant is as shown on the left. The first derivative for 1 is 0, for x it is 1, and for x^2 it is 2x. The second derivatives respectively are 0, 0, 2.

Plug in the values for the first and second derivatives found in step two into the determinant. The Wronskian is 1 * (1*2 -- 0) - 0 + 0. Thus W(1, x, x^2) = 2. Since this is never 0, the three functions are linearly independent.

#### References

- Differential Equations; Shepley L. Ross; 1984
- Mathematical Methods in the Physical Sciences; Mary Boas; 2005

#### Resources

#### Photo Credits

- Wikimedia Commons