8. Partial Differential Equations

Try these first

1. 1.

   Define a Partial Differential Equation (PDE).

   Hint

   Think about what variables and derivatives are involved.

2. 2.

   What distinguishes a linear PDE?

   Hint

   Recall the characteristics of linear equations.

3. 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

   Hint

   Recall the properties of homogeneous equations.

4. 4.

   True or False: In a homogeneous linear PDE, a free term may be present.

   • True
   • False

   Hint

   Think about the definition of homogeneity.

5. 5.

   Solve the PDE ∂²z/∂x² + 3∂²z/∂y² = 0 and interpret the nature of the roots. What kind of solutions would you expect?

   Hint

   Focus on how each term in the operator form translates to the auxiliary equation.

6. 6.

   For the equation ∂²z/∂x² - 6∂²z/∂x∂y + 9∂²z/∂y² = 0, find the general solution and describe the form it takes.

   Hint

   Recognize how repeated roots influence your complementary function.