Practice Partial Differential Equations (8) - Homogeneous Linear PDEs with Constant Coefficients

Practice: Partial Differential Equations

Lesson 8Homogeneous Linear PDEs with Constant Coefficients

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8 questions

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Exercise
6 questions · about 4 min · pass mark 70%. Hints available.
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2 questions · about 1 min · pass mark 70%. Hints available.
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6 sample questions

Question 1.
Define a Partial Differential Equation (PDE).
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Think about what variables and derivatives are involved.
Question 2.
What distinguishes a linear PDE?
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Recall the characteristics of linear equations.
3Question 3.
What is the general form of a homogeneous linear PDE?
- a) ax + by + c = 0
- b) ∂²z/∂x² + ∂²z/∂y² = 0
- c) ∂²z + 5x = 0
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Recall the properties of homogeneous equations.
4Question 4.
True or False: In a homogeneous linear PDE, a free term may be present.
- True
- False
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Think about the definition of homogeneity.
Question 5.
Solve the PDE ∂²z/∂x² + 3∂²z/∂y² = 0 and interpret the nature of the roots. What kind of solutions would you expect?
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Focus on how each term in the operator form translates to the auxiliary equation.
Question 6.
For the equation ∂²z/∂x² - 6∂²z/∂x∂y + 9∂²z/∂y² = 0, find the general solution and describe the form it takes.
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Recognize how repeated roots influence your complementary function.

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