Page 242 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 16: Classifications of Integrals of the First Order Partial Differential Equations

Eliminating the constant c and we get the equation (4). Notes
We see that the common things among these two surfaces of revolution (1) and (5) is that they
have the line OZ as the axis of symmetry. So if we simply take the equation

2
2
z = f(x + y ) ...(7)
where the function f is arbitrary and again differentiate (7) with respect to x and y separately we
get
z z
p 2xf , 2yf ...(8)
x y

f
2
where f and u = x + y . So after eliminating f from (8)
2
u
we get py qx = 0 ...(4)
Thus we see that the function z defined by each of the equations (1), (5) and (7), is in some sense
a solution of the equation.

We now interpret the argument slightly. The relation (1) and (5) are both of the type
F(x, y, z, a, b) = 0 ...(9)
where a and b denote arbitrary constants. If we differentiate this equation with respect to x and
y respectively. We obtain the relations

F F F F
p 0, q 0
x z y z ...(10)
The set of equations (9) and (10) constitute three equations involving two arbitrary constants
a and b. It will be possible to eliminate a and b from these equations to obtain a relation of the
kind
f(x, y, z, p, q) = 0 ...(11)
showing that the system of surfaces gives rise to a partial differential equation (11) of the first
order.
The obvious generalization of the equation (7) is a relation between x, y, z of the type
F(u, v) = 0 ...(12)

where u and v are functions of x, y and z and F is an arbitrary function of u and v. If we
differentiate (12) with respect to x and y respectively, we obtain the relations

F u u F v
p p 0
u x z dv x z

F u u F v v
q q 0
u y z dv y z

F  F
and if we eliminate and from these equations, we obtain the equation
u dv

F u u v v v v u u
p q p q 0
u x z y z x z y y

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