Question 4.8.4: Testing a Set of Solutions for Linear Independence Determine...
Testing a Set of Solutions
for Linear Independence
Determine whether {e^x, xe^x, (x + 1)e^x} is a set of linearly independent solutions of the linear homogeneous differential equation
y″′ – 3y″ + 3y′ – y = 0.
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As in Example 3, begin by verifying that each of the functions is actually a solution of y″′ – 3y″ + 3y′ – y = 0. (This verification is left to you.) Testing for linear independence produces the Wronskian of the three functions, as follows.
W = \left | \begin{matrix} e^x & xe^x & (x + 1)e^x \\ e^x & (x + 1)e^x & (x + 2)e^x \\ e^x & (x + 2)e^x & (x + 3)e^x \end{matrix} \right | = 0
So, the set {e^x, xe^x, (x + 1)e^x} is linearly dependent.
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