Let us find the Wronskian of the following functions and determine whether it is linearly dependent or linearly independent on (−∞,∞).
1. {x2,x+1,x−3}
Since
W(x2,x+1,x−3)=∣∣x22x2x+110x−310∣∣=2(x+1)−2(x−3)=2x+2−2x+6=8=0,
we conclude that the functions are linearly independent.
2. {3e2x,e2x}
Since
W(3e2x,e2x)=∣∣3e2x6e2xe2x2e2x∣∣=6e4x−6e4x=0,
we conclude that the functions are linearly dependent.
3. {x2,x3,x4}
Since
W(x2,x3,x4)=∣∣x22x2x33x26xx44x312x2∣∣=36x6+8x6+12x6−6x6−24x6−24x6=2x6=0,
we conclude that the functions are linearly independent.
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