1. 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

2. 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

3. 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

4. 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:

5. 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

6. 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