Let f(x,y,z,ux,uy,uz)=0 (1)
The Fundamental idea of the Jacobi method is to introduce two first order PDE involving two arbitrary constants a and b of the following form,
h1(x,y,z,ux,uy,uz,a)=0 (2)
h2(x,y,z,ux,uy,uz,b)=0 (3)
such that ∂(ux,uy,uz)∂(f,h1,h2)=0 (4)
and
Equations (1),(2) and (3) can be solved for ux,uy,uz
and equation du=uxdx+uydy+uzdz is integrable
If h1=0 and h2=0 are compitable withf=0 then h1,h2 satisfy,
∂(x,ux)∂(f,h)+∂(y,uy)∂(f,h)+∂(z,uz)∂(f,h)=0 for h=hi, i =1,2
Last equation will lead to the PDE of the form,
fux∂x∂h+fuy∂y∂h+fuz∂z∂h−fx∂ux∂h−fy∂uy∂h−fz∂uz∂h=0
It's auxiliary equation will be,
fuxdx=fuydy=fuzdz=−fxdux=−fyduy=−fzduz
After getting this we need to those fractions which will lead to the solution.
Since given equation is
f(x,ux,uz)=g(y,uy,uz)
it can be simplified as
f(x,ux,uz)−g(y,uy,uz)=0
or it can be written as
F(x,y,ux,uy,uz)=0
Then auxiliary equation will be
Fuxdx=Fuydy=Fuzdx=−Fxdux=−Fyduy
Solve it to obtain the solution.
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