10.5. General Form of Solvable PDEs

Try these first

1. 1.

   What is a Partial Differential Equation (PDE)?

   Hint

   Look at the definition we discussed in class.

2. 2.

   What does direct integration involve?

   Hint

   Consider how we solve simple equations.

3. 3.

   Which of the following is a first-order PDE format?

   • ∂²z/∂y² = g(y)
   • ∂z/∂x = h(x)
   • ∂³z/∂x³ = f(x)

   Hint

   Focus on the order of derivatives.

4. 4.

   True or False: Arbitrary functions are never included in the solutions of PDEs.

   • True
   • False

   Hint

   Consider the integration process along with our discussions.

5. 5.

   Solve the following system of PDEs: ∂z/∂x = x + y and ∂z/∂y = 2x - y.

   Hint

   Make sure to relate your two derived functions.

6. 6.

   Given ∂²z/∂x² + ∂²z/∂y² = 0, solve using direct integration by separating variables.

   Hint

   Recap the role of each variable in isolation when integrating.