Question

[C] Find the Wronskian of the following functions and determine whether it is linearly dependent or linearly independent on (-∞,∞).

1. {x², x+1, x-3}               ans, W=8, linearly independent
2. {3e^2x, e^2x}                     ans, W=0, linearly dependent
3. {x², x³, x⁴}                    ans, W=2x^6, linearly independent

Answer 1

Let us find the Wronskian of the following functions and determine whether it is linearly dependent or linearly independent on $(-∞,∞).$

1.   $\{x^2, x+1, x-3\}$

Since

$W(x^2, x+1, x-3)= \begin{vmatrix} x^2 & x+1 & x-3\\ 2x & 1 & 1\\ 2 & 0 & 0 \end{vmatrix} \\=2(x+1)-2(x-3)=2x+2-2x+6=8\ne 0,$

we conclude that the functions are  linearly independent.

2.   $\{3e^{2x}, e^{2x}\}$

         Since

$W(3e^{2x}, e^{2x})= \begin{vmatrix} 3e^{2x} & e^{2x}\\ 6e^{2x} & 2e^{2x} \end{vmatrix} =6e^{4x}-6e^{4x}= 0,$

we conclude that the functions are  linearly dependent.   

 

3.   $\{x^2, x^3, x^4\}$

        Since

$W(x^2, x^3, x^4)= \begin{vmatrix} x^2 & x^3 & x^4\\ 2x & 3x^2 & 4x^3\\ 2 & 6x & 12x^2 \end{vmatrix} \\=36x^6+8x^6+12x^6-6x^6-24x^6-24x^6=2x^6\ne 0,$

we conclude that the functions are  linearly independent.