4.3. Linear First-Order PDEs: Lagrange’s Method

Try these first

1. 1.

   What is a first-order PDE?

   Hint

   Think about the order of derivatives.

2. 2.

   Write the standard form of a first-order linear PDE.

   Hint

   Recall P, Q, and R represent the coefficients.

3. 3.

   Which of the following is the correct form of a first-order linear PDE?

   • Pz + Qy = R
   • P(x,y,z)p + Q(x,y,z)q = R(x,y,z)
   • d/dx(P) + d/dy(Q) = R

   Hint

   Pay attention to the variables involved in the equation.

4. 4.

   True or False: Auxiliary equations for Lagrange’s method come from the solution of the PDE itself.

   • True
   • False

   Hint

   Think about how we arrive at the auxiliary equations.

5. 5.

   Prove the effectiveness of Lagrange’s method by solving a non-homogeneous first-order linear PDE such as z + 2x = y.

   Hint

   Start with the ratios from the coefficients.

6. 6.

   Explore the limits of Lagrange's method by attempting to apply it to a non-linear PDE and discuss the Schrodinger analogy.

   Hint

   Focus on the non-linearity aspect when setting up your approach.