Solutions to Partial Differential Equations Via Lagrange Method

 1)  Find two integrals (solutions) for:

dx/z²    =  dy/ x( z - y) = dz/ x

           Solution:

Treat two equations:

A)  dx/z²    =  dz/ x

B)  dy/ x( z - y) = dz/ x

For (A)  rewrite, separating variables:

x dx =     z² dz

 Then the solution from integration is:

½   x² =   1/3  (z ³ ) + c1

For (B),  note that a multiplication by x yields an equation containing only y and z:

dy =  (z - y) dz

Or:

dy/ dz   + y  = z   

Which is a 1st order linear equation with solution:

y = z - 1  + c2 exp (-z)

Then we arrive at the two final results:   

e ^z (y + 1 - z)  = c2

and:

3x²  -   2 z ³ = c3

 

2)  Find two integrals of:

dx/x   =  dy/ y  = dz/ xy

  Solution:

Note the first equation: dx/x   =  dy/ y    has for its solution:

y = c1 x

Substitute this solution into the denominator of dz to obtain:

dx/x   =  dz/ c1x²

or:  c1x dx = dz

Integration yields:

½ c1x²    =   z  + c2

Or:  2z    =  yx + c2

Then we obtain:

y = c1 x    and 2z    =  yx + c2