9.4. Example Problems

Try these first

1. 1.

   What is a Non-Homogeneous Linear PDE?

   Hint

   Think about what the right side of the equation signifies.

2. 2.

   Define the term Complementary Function.

   Hint

   What solution do we find from the homogeneous equation?

3. 3.

   What is the term for a solution to the homogeneous part of a PDE?

   • Particular Integral
   • Complementary Function
   • Standard Form

   Hint

   Think about what 'homogeneous' means in mathematics.

4. 4.

   True or False: A non-homogeneous PDE has a zero function on the right side.

   • True
   • False

   Hint

   Consider the definition of non-homogeneous.

5. 5.

   Solve the equation (D² - D'²)z = e^{2x} cos(3y). Determine both CF and PI.

   Hint

   Carefully analyze how the non-homogeneous part informs your assumed solution form.

6. 6.

   For the PDE (D² + D'²)z = x^3y^2, derive the general solution, showing all steps.

   Hint

   Focus on the patterns presented by polynomial right sides, and ensure your assumptions reflect that.