31.13 - Similarity and Systems of Linear Differential Equations
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Practice Questions
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What is the form of a linear system of ODEs?
💡 Hint: Recall the basic structure of ODEs.
Define a diagonalizable matrix.
💡 Hint: Think about the eigenvalue representation.
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Interactive Quizzes
Quick quizzes to reinforce your learning
What form does a linear system of ODEs take?
💡 Hint: Think about the standard representation in ODEs.
Is a matrix that cannot be diagonalized still useful?
💡 Hint: Consider how we might express systems even without diagonalization.
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Challenge Problems
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Given a matrix A = [[3, 1], [0, 3]], determine its eigenvalues and discuss whether the system of ODEs it describes can be simplified using diagonalization.
💡 Hint: Review the characteristic polynomial for solutions.
Consider a linear system represented by dx/dt = A for A = [[0, 1], [-1, 0]]. Describe how this might represent a physical system and analyze its behavior over time by finding its eigenvalues.
💡 Hint: Use the determinant method to find eigenvalues.
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