5.5. Solved Examples

Try these first

1. 1.

   Identify P, Q, and R in the equation ∂z/∂x + 2∂z/∂y = 3z.

   Hint

   Express the PDE in the standard form Pp + Qq = R.

2. 2.

   What is the general form of a first-order PDE?

   Hint

   Recall the elements of the equation.

3. 3.

   What is the form of Lagrange's Linear Equation?

   • Pp + Qq = 0
   • Pp + Qq = R
   • ∂z/∂x + ∂z/∂y = z

   Hint

   Remember the equation form we discussed in class.

4. 4.

   Auxiliary equations arise from what kind of transformations?

   • True
   • False

   Hint

   Think about how we derived them in examples.

5. 5.

   Given the PDE ∂z/∂x + ∂z/∂y = 5, use Lagrange's method to find the general solution and explain the steps.

   Hint

   Pay attention to the integration steps and constants involved.

6. 6.

   For the equation ∂z/∂x + (sqrt(x) + 1)∂z/∂y = 0, derive the general solution using Lagrange's characteristics.

   Hint

   Notice how sqrt(x) influences the auxiliary equation structure.