Using Jacobi's method find the complete integral of the equation
2u1xz+3u2z2+u22u3=0This equation is already in the form of
F(x,y,z,u1,u2,u3)=0 Its auxiliary equations are
Fu1dx=Fu2dy=Fu3dz=−Fxdu1=−Fydu2=−Fzdu3
2xzdx=3z2+2u2u3dy=u22dz=−2u1zdu1=0du2=−2u1x−6u2zdu3 From the first and fourth ratios, we get
2xzdx=−2u1zdu1=>−xdx=u1du1=>lnx+lnu1=lnau1x=a,a=const From the fifth ratio, we get
0du2=>du2=0=>u2=b,b=const With the above forms of u1 and u2, we obtain from the given equation
2az+3bz2+b2u3=0u3=−b22az+3bz2The equation
du=u1dx+u2dy+u3dz obtained for these values of u1,u2,u3, is integrable.
du=(xa)dx+bdy+(−b22az+3bz2)dz On integrating the above equation, we arrive at the following solution of the given equation
u=alnx+by−b2az2−bz3+c,a=const,b=const,c=const
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