4.3.2. Lagrange’s Auxiliary Equations

Try these first

1. 1.

   Define a first-order PDE in your own words.

   Hint

   Focus on the term 'first derivative' in your definition.

2. 2.

   What are auxiliary equations used for?

   Hint

   Think about how we relate differentials to one another.

3. 3.

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

   • F(x
   • y
   • z
   • p
   • q) = 0
   • P(x
   • y
   • z)p + Q(x
   • y
   • z)q = R(x
   • y
   • z)
   • ∂z/∂x + ∂z/∂y = 0

   Hint

   Look for the setup that mentions P, Q, and R.

4. 4.

   True or False: The auxiliary equations are used to derive two independent solutions.

   • True
   • False

   Hint

   Think about how we apply these equations.

5. 5.

   Given the PDE pz + 2qy = 3x, set up the auxiliary equations and find the general solution.

   Hint

   Focus on integrating the ratios correctly.

6. 6.

   Explain how varying the auxiliary equations impacts the outcome of the general solution.

   Hint

   Consider the role of different inputs into the same formula.