Practice Example Problems (8.4) - Homogeneous Linear PDEs with Constant Coefficients

Practice: Example Problems

Lesson 8.4Homogeneous Linear PDEs with Constant Coefficients

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9 questions

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Exercise
6 questions · about 4 min · pass mark 70%. Hints available.
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3 questions · about 2 min · pass mark 70%. Hints available.
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6 sample questions

Question 1.
Convert the PDE $\frac{\partial^2 z}{\partial x^2} - 4 \frac{\partial^2 z}{\partial y^2} = 0$ to operator form.
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Replace the derivatives with their operator equivalents.
Question 2.
Define what a homogeneous PDE is in your own words.
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Think about the role of terms in the equation.
3Question 3.
What type of PDE has no free terms?
- Homogeneous
- Non-homogeneous
- Linear
- Non-linear
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Remember the definition of homogeneous equations.
4Question 4.
True or False: A repeated root leads to a general solution involving both $f(y - mx)$ and $x f(y - mx)$ .
- True
- False
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Recall how the roots affect the general solution forms.
Question 5.
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.
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Analyze the discriminant of the auxiliary equation to categorize the roots.
Question 6.
Create your own PDE that is homogeneous and has a distinct root, then solve it and present your general solution.
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Ensure your constants create a distinct quadratic equation for compliance.

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