University Mathematics Handbook (2015)

XIV. Partial Differential Equations (PDE)

Chapter 2. First-Order Linear PDE

 (*)

when is an unknown function piecewise differentiable by and , functions , , , are continuous in and , are not vanished together.

2.1  General Solution

Finding the solutions systematically:

a. Construct a characteristic equation

b. The characteristic equation's solutions are called characteristic curves of (*)

c. Choose arbitrary function of class such that holds

d. Changing variables , in (*), we get a partially differential equation with respect to : whose solution is .

e. General solution of (*) is .

2.2  Cauchy Problem

a.  Cauchy problem for a PDE consists of equation (*) and initial condition

 (**)

b.  If initial condition (**) is such that for every vectors and are not parallel, then Cauchy problem has a unique integral surface (see 3.2.b).