Question #281746

[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. {3e2x, e2x}                     ans, W=0, linearly dependent
  3. {x2, x3, x4}                    ans, W=2x^6, linearly independent
1
Expert's answer
2021-12-26T17:41:45-0500

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


1.   {x2,x+1,x3}\{x^2, x+1, x-3\}

Since


W(x2,x+1,x3)=x2x+1x32x11200=2(x+1)2(x3)=2x+22x+6=80,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.   {3e2x,e2x}\{3e^{2x}, e^{2x}\}

         Since


W(3e2x,e2x)=3e2xe2x6e2x2e2x=6e4x6e4x=0,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.   {x2,x3,x4}\{x^2, x^3, x^4\}

        Since


W(x2,x3,x4)=x2x3x42x3x24x326x12x2=36x6+8x6+12x66x624x624x6=2x60,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.          


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