Chapter 1:First order Partial Differential Equations Sec 1.2: The Linear Equation Consider the linear first order partial differential equation in two independent variables: We assume that a, b, c, and f are functions in (x,y) . They are continuous in some region of the plane. a(x,y) and b(x,y) are not both zero for the same (x,y) (???) We will show how to solve this equation. The key is to determine a change of variable

Chapter 1:First order Partial Differential Equations Sec 1.2: The Linear Equation PDE ODE Consider the ODE: 1st order linear ODE Method: Find integrating factor = K Multiply equation by K LHS = Integrate both sides

Chapter 1:First order Partial Differential Equations Sec 1.2: The Linear Equation Characteristic equation Consider the linear 1st PDE Find the general solution of thePDE solution

Chapter 1:First order Partial Differential Equations Sec 1.2: The Linear Equation Consider the linear first order partial differential equation in two independent variables: Find the characteristic equation: Find the general solution of the characteristic equation and put it in the form: Use the transformation: To change PDE into this form:

1 2 3 4 Chapter 1:First order Partial Differential Equations Sec 1.2: The Linear Equation Consider the linear 1st PDE 1 characteristic equation: 2 Solution: 3 transformation: 4

1 2 3 4 Chapter 1:First order Partial Differential Equations Sec 1.2: The Linear Equation Consider the linear 1st PDE characteristic curves: 1 characteristic equation: C=4 2 Solution: C=1 3 transformation: C=-4 4