How to calculate

Linear Independence

If $W(f, g)(x_0) \ne 0$ for some $x_0$, then $f(x)$ and $g(x)$ are linearly independent. If $W(f, g)(x_0)=0$ for some $x_0$, then $f(x)$ and $g(x)$​ are linearly dependent.

For a differential equation, $y_1$ and $y_2$ must be linearly independent in order to be solutions to the DE

Example

Problem 8: Determine if the following functions are linearly independent or dependent: $f(t)=2t^2$ and $g(t)=t^4$​

Solution 8: Use the Wronskian to determine linear independence

As long as $t \ne 0$, then $f(t)$ and $g(t)$ are linearly independent.

Code

We use sympy for this, since we are dealing with symbolic mathematics.

import sympy
from sympy import *
x, y, z, t = symbols('x y z t')

Function to calculate wronskian below:

def wronskian(f, g):
  fp = diff(f, t)
  gp = diff(g, t)
  wronskianm = Matrix([[f, g], [fp, gp]])
  return wronskianm.det()

Testing the function on the example we solved above.

f = 2*t**2
g = t**4
wronskian(f, g)

[OUTPUT]
4*t**5

4*t**5 is equivalent to $4t^5$.

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Wikipedia Article