23.11 - The Wronskian and Linear Independence of Functions
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Practice Questions
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What is the Wronskian of the functions f(x) = e^x and g(x) = e^(2x)?
💡 Hint: Recall the structured form of the Wronskian determinant.
Define linear independence in the context of functions.
💡 Hint: Think about what it means for vectors in a vector space!
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Interactive Quizzes
Quick quizzes to reinforce your learning
What does a non-zero Wronskian indicate about a set of functions?
💡 Hint: Remember the significance of the Wronskian!
True or False: The functions sin(x) and cos(x) have a Wronskian that is always zero.
💡 Hint: Think back to how large the functions can be differentiated.
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Challenge Problems
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Calculate the Wronskian for the functions f1(x) = ln(x), f2(x) = x^2, and check for linear independence.
💡 Hint: Remember to differentiate the functions before forming the Wronskian!
Provide a real-world scenario where linear independence of solutions is necessary. Discuss the implications of dependent vs. independent solutions.
💡 Hint: Discuss the importance of independent forces in structural integrity.
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