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Fundamentals Of Differential Equations And Boundary Value Problems
Found in: Page 326
Fundamentals Of Differential Equations And Boundary Value Problems

Fundamentals Of Differential Equations And Boundary Value Problems

Book edition 9th
Author(s) R. Kent Nagle, Edward B. Saff, Arthur David Snider
Pages 616 pages
ISBN 9780321977069

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Short Answer

Determine whether the given functions are linearly dependent or linearly independent on the specified interval. Justify your decisions.

{cos2x,cos2x,sin2x} on (-,)

Therefore, the function cos2x,cos2x,sin2x is linearly dependent on -,.

See the step by step solution

Step by Step Solution

Step 1: Using the concept of Wronskian

The given function is cos2x,cos2x,sin2x.

Apply the concept of Wronskian,

Wf1,f2,,fn=f1xf2xfnxf1'xf2'xfn'xf1n-1xf2n-1xfnn-1x

Therefore,

Wcos2x,cos2x,sin2x=Wcos2x,1+cos2x2,1-cos2x2=cos2x1+cos2x21-cos2x2-2sin2x-sin2xsin2x-4cos2x-2cos2x2cos2x

Solve the above equation,

Wcos2x,cos2x,sin2x=cos2x-2sin2xcos2x+2sin2xcos2x-1+cos2x2-4sin2xcos2x+4sin2xcos2x+1-cos2x24sin2xcos2x-4sin2xcos2x=cos2x0-1+cos2x20+1-cos2x20=0

Step 2:Check the linearly independent or dependent

The above function is equal to zero x.

Therefore, cos2x,cos2x,sin2x is linearly dependent on -,.

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