Page 232 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 15: Cauchy’s Problem and Characteristics for First Order Equations

At this point we want to say a few words about different kinds of solutions. We may get a Notes
relation of the type
F(x, y, z, a, b) = 0

for the solution of the first order partial differential equation.
Any such relation containing two arbitrary constants a and b and a solution of the partial
differential equation of the first order is said to be a complete solution or a complete integral of
that equation.
On the other hand any relation of the type
F(u, v) = 0

involving an arbitrary function F connecting two known functions u and v of x, y and z and
providing a solution of the first order partial differential equation is called a general solution or
a general integral of that equation.
We shall be dealing with the classifications of the integrals of the first order partial differential
equations in the unit 16 in more details.

Self Assessment

1. Eliminate constants a and b from the equation

z = (x + a) (y + b)
2. Eliminate the arbitrary function f from the equation
2
2
z = xy + f (x + y )
15.2 Cauchy’s Method of Characteristics

We should now consider a method due to Cauchy for solving the non-linear partial differential
equation

z z
F(x, y, z, , ) = 0 ...(1)
x y
The method is based on geometrical ideas. Equation (1) can be theoretically solved to obtain an
expression.
q = G (x, y, z, p) ...(2)
from which q is calculated in terms of x, y, z and p. Before proceeding further let us consider a
plane passing through a point P(x , y , z ) with its normal parallel to the direction n defined by
0 0 0
the direction cosines (p , q , 1). This plane is uniquely specified by the set of numbers D(x , y ,
0 0 0 0
z , p , q ). Conversely any such set of five numbers defines a plane in three dimensional space.
0 0 0
We now define
A plane element: A set of five numbers D(x, y, z, p, q) is called a plane element of the space.
An integral element: If the plane element (x, y, z, p, q) satisfies an equation
F(x, y, z, p, q) = 0 ...(3)
it is called an integral element of the equation (3) at the point (x , y , z ).
0 0 0
Thus keeping x , y and z fixed and varying p, we obtain a set of plane elements {x , y , z , p,
0 0 0 0 0 0
G(x , y , z , p)} which depend on the single parameter p. As p varies we obtain a set of plane
0 0 0

LOVELY PROFESSIONAL UNIVERSITY 225