For general linear partial differential equations of the first order we may write, in x, y and z:
1) P(x, y, z) z x + Q(x, y, z) z y = R(x, y, z)
the technique for finding the general solution of this equation is to find wo integrals of the system of ordinary differential equations:
2) dx/ P(x,y,z) = dy / Q (x,y,z) = dz/ R(x,y,z)
Now, let u(x,y,z) = a and v(x,y,z) = b, Then either:
3) f (u, v) = 0 or: u = y(v)
Is a general solution of (1):
Example Problem: Find the general solution of:
x2 z x - xy zy = y2
Solution:
Form the system of ordinary differential equations:
dx/ P = dy/ Q = dz/ R
Or:
dx/x2 = dy/ -xy = dz/ y2
Two independent solutions of the system are desired. To obtain the 1st:
dx/x2 = dy/ -xy Or: dx/ x + dy/ y = 0
Which yields: xy = c1
The 2nd solution is obtained by using the first solution in the equation of the last two fractions, so:
dy/ -c1 = dz/ y2 Or: y2 dy = - c1 dx
Which has the solution:
y3 /3 + c1 z = c2
Since c1 = xy:
y2 + 3 xz = 3 c2 y-1 = c3 x
Since c3 is an arbitrary constant we can let it be an arbitrary function of c1, f (c1)
then replace c1 by c1 = xy to obtain the general solution:
y2 + 3 xz = x f (xy)
This technique is known as the method of Lagrange, after the 18th century French mathematician Joseph-Louis Lagrange. It is based on the fact that any solution: u(x,y,z) = c of the system:
dx/ P = dy/ Q = dz/ R
Is also a solution of:
P ( ¶ u/ ¶ x ) + Q ( ¶ u/ ¶ y ) + R ( ¶ u/ ¶ y ) = 0
Suggested Problems For The Math Whiz:
1) Find two integrals (solutions) for:
dx/z2 = dy/ x( z - y) = dz/ x
2) Find two integrals of:
dx/x = dy/ y = dz/ xy
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