Page 244 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

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

Notes
dz x xy
and q = f 2 q ...(16)
dy z z
p yz xyp
so =
q xz xyq
or pxz xypq = yzq xypq

or z(px qy) = 0
is the answer.

Self Assessment

1. Eliminate the constants a and b from the equation
2
2
2
ax + by + z = 1
2. Eliminate the arbitrary function from the equation
2
2
2
2
F(x + y + z , z 2xy) = 0
16.2 Classes of Integrals of a Partial Differential Equation
Let us consider the partial differential equation of the form
F(x, y, z, p, q) = 0 ...(1)

in which the function F is not necessarily linear in p and q. We saw earlier that the solution
involving two parameter system of equation can be of the form
f(x, y, z, a, b) = 0 ...(2)

Any envelope of the system (2) must also be a solution of the differential equation (1). In this
way we are led to three classes of integrals of a partial differential equation of type (1):
(a) Two parameter systems of surfaces f(x, y, z, a, b) = 0.

Such an integral is called complete integral.
(b) If we take any one parameter subsystem
f(x, y, z, a, (a)) = 0
of the system (2) and form its envelope, we obtain a solution of equation (1). When the
function (a) which defines the subsystem is arbitrary, the solution obtained is called
general integral of (1) corresponding to the complete integral (2).

When a definite function (a) is used we obtain a particular case of the general integral.
(c) If the envelope of the two parameter system (2) exists, it is also a solution of the equation
(1), it is called the singular integral of the equation.

Example 1: Show that
2
z = ax + by + a + b 2 ...(1)
is the complete integral of partial differential equation
2
z = px + qy + p + q 2 ...(2)

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