Mathematical Physics Vol 1

7.3 Linear and quasilinear first order PDE

351

General solution If the parameters a and b are not independent, that is, if, for example, b = b ( a ) , then the surface envelope is defined by the relation

∂ g ∂ a

∂ g ∂ b

d b d a

g ( x , y , z , a , b ( a ))= 0 ,

= 0 .

(7.57)

+

This solution is called the general solution (general integral).

R Note. If one complete solution of equation (7.52) is known, a general and singular solu tion (if it exists) can be obtained from it, by simply differentiating and eliminating the appropriate parameters. Let us demonstrate this claim. Assume that the complete integral of the equation (7.52) is the function (7.53) g ( x , y , z , a , b )= 0 . (7.58) Let parameters a and b be functions of x and y . Then, differentiating (7.58) first by x , and then by y , we obtain

∂ g ∂ x ∂ g ∂ y

∂ g ∂ z ∂ g ∂ z

∂ g ∂ a ∂ g ∂ a

∂ a ∂ x ∂ a ∂ y

∂ g ∂ b ∂ g ∂ b

∂ b ∂ x ∂ b ∂ y

p +

= 0 ,

+

+

(7.59)

q +

= 0 .

+

+

Parameters a and b are determined from the condition ∂ g ∂ a ∂ a ∂ x + ∂ g ∂ b ∂ b ∂ x

= 0 ,

(7.60)

∂ g ∂ a

∂ a ∂ y

∂ g ∂ b

∂ b ∂ y

= 0 .

+

From this homogeneous system of equations we can determine ∂ g ∂ a and

∂ g ∂ b

. Then we have

two different cases: • the system determinant is not equal to zero, i.e.

∂ a ∂ x ∂ a ∂ y

∂ b ∂ x ∂ b ∂ y ̸

= 0 ,

and the system has only trivial solutions ∂ g ∂ a =

∂ g ∂ b

= 0 .

(7.61)

From these equations we determine a and b as functions of x and y andobtain g = g ( x , y , z , a ( x , y ) , b ( x , y )) ,

which represents the singular solution. • The system determinant is equal to zero

∂ a ∂ x ∂ a ∂ y

∂ b ∂ x ∂ b ∂ y

= 0 ,

from where two possibilities follow