Page 235 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes
pdx qdq pdx qdy dz
Also F = F F F F F
p q p q p q
p p p p p p

dx dy dz
Hence = ...(13)
F F F F
p q
p p p p
that means that along a characteristic strip, x (t), y (t), z (t) must be proportional to F , F , p F + q
p q p
F respectively. If we choose the parameter t in such a way that
q
x (t) = F , y (t) = F ...(14)
p q
then z (t) = p F + q F
p q
along a characteristic strip p is a function of t so that

p p
t
t
p (t) = x ( ) y ( )
x y
p F p F
=
x p y q
p F q F Since p q
= . y x
x p x q

Differentiating equation (3) with respect to x, we find that
F F F p F q
p + = 0
x z p x q x

so that on a characteristic strip
p (t) = (F + p F ) ...(16)
x z
and it can be shown similarly that
q (t) = (F + q F ) ...(17)
y z
Collecting equations (14) to (17), we see that we have the following system of five ordinary
differential equations for the determination of the characteristic strip
x (t) = F , y (t) = F , z (t) = p F + q F q
p q p
p (t) = (F + p F ), q (t) = (F + q F ) ...(18)
x z y z
These equations are known as the characteristic equations of the differential equation (3).
The main theorem about characteristic strip is:

Theorem 2: Along every characteristic strip of the equation F(x, y, z, p, q) = 0, the function F(x,
y, z, p, q) is a constant.
The proof is a matter simply of calculation. Along a characteristic strip we have

d
t
p
t
t
z
q
t
x
F ( ( ), ( ), ( ), ( ), ( )) F x F y F z F p F q
y
t
z
y
x
q
p
dt
F F p F F q F z (pF p qF q ) F p (F x pF z ) F q (F y qF z ) 0
y
x
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