Wednesday, January 29, 2025

Solution of First Order Partial Differential Equations (The Method Of Lagrange)

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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