Page 252 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 17: Lagrange’s Methods for Solving Partial Differential Equations

Notes

Note: If generalised to n independent variables, obviously the equation is

P p + P p + P p + ... + P p = R ... (2)
1 2 2 2 3 3 n n
where P , P , ... P , R are functions of n independent variables x , x , ... x and a dependent variable
1 2 n 1 2 n
f
f; p = , (i = 1, 2, ... n).
i
x i
It should be noted that the term ‘linear’ in the section means that p and q (or, in general case p ,
1
... p ) appear to the first degree only, but P, Q and R may be any functions of x, y and z.
n
17.2 Lagrange’s Method of Solutions

The Lagrange’s equation is
Pp + Qq = R ... (1)
where P, Q, R are functions of x, y, z. Suppose
u = f(x, y, z) = a ... (2)

is a relation that satisfies (1). Differentiating (2) with respect to x, y,
u u z
= 0,
x z x

u u z
And = 0
y z y
u u
or p = 0
x z
u u
and q = 0
y z

u u
x y
Hence p = and q
u u
z z

Substituting these values of p and q in (1) changes it to
u u u
P Q R = 0 ... (2)
x y z

Therefore, if u = a be an integral of (1), u = a also satisfies (2). Conversely if u = a be an integral
u
of (2), it is also an integral of (1). This can be seen by dividing by and substituting p and q for
z
the values above. Therefore equation (2) can be taken as equivalent to equation (1).

We have shown in unit (8) that u = a and v = b are independent solution of the system of equations
dx dy dz
... (3)
P Q R

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