16.10 - Canonical (Standard) Forms of Second-Order PDEs
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Practice Questions
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What is the general form of a second-order PDE?
💡 Hint: Look for the highest order derivatives present.
Define discriminant in the context of second-order PDEs.
💡 Hint: Recall how it relates to the coefficients A, B, and C.
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Interactive Quizzes
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What is the canonical form of an elliptic PDE?
💡 Hint: Recall the different forms discussed in class.
If D = B^2 - 4AC < 0, which type of PDE do you have?
💡 Hint: Think about how discriminants categorize the equations.
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Challenge Problems
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Given the PDE \( 2 \frac{\partial^2 u}{\partial x^2} - 8 \frac{\partial^2 u}{\partial x \partial y} + 4 \frac{\partial^2 u}{\partial y^2} = 0 \), classify it and find the canonical form.
💡 Hint: Pay close attention to the coefficients A, B, and C.
Derive a transformation for the equation \( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0 \) into its canonical form.
💡 Hint: Focus on recognizing how changes in variables can simplify complex equations.
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