8.4 - Example Problems
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Practice Questions
Test your understanding with targeted questions
Convert the PDE \( \frac{\partial^2 z}{\partial x^2} - 4 \frac{\partial^2 z}{\partial y^2} = 0 \) to operator form.
💡 Hint: Replace the derivatives with their operator equivalents.
Define what a homogeneous PDE is in your own words.
💡 Hint: Think about the role of terms in the equation.
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Interactive Quizzes
Quick quizzes to reinforce your learning
What type of PDE has no free terms?
💡 Hint: Remember the definition of homogeneous equations.
True or False: A repeated root leads to a general solution involving both \(f(y - mx)\) and \(x f(y - mx)\).
💡 Hint: Recall how the roots affect the general solution forms.
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Challenge Problems
Push your limits with advanced challenges
Consider the PDE \( \frac{\partial^2 z}{\partial x^2} + k \frac{\partial^2 z}{\partial y^2} + m \frac{\partial z}{\partial x \partial y} = 0 \) where k and m are constants. Derive the general solution when k > 0 and m < 0.
💡 Hint: Analyze the discriminant of the auxiliary equation to categorize the roots.
Create your own PDE that is homogeneous and has a distinct root, then solve it and present your general solution.
💡 Hint: Ensure your constants create a distinct quadratic equation for compliance.
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