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Question
If u =`f((y-x)/(xy),(z-x)/(xz)),` show that `x^2(delu)/(delx)+y^2(delu)/(dely)+z^2(delu)/(delz)=0`.
Solution
let `u=f(r,s)`
`thereforer=(y-x)/(xy) therefores=(z-x)/(xz)`
`therefore (delu)/(delx)=(delu)/(delr)(delr)/(delx)+(delu)/(dels)(dels)/(delx)=(delu)/(delr)1/(x^2)+(delu)/(dels)((-1)/x^2)`
`(delu)/(dely)=(delu)/(delr)(delr)/(dely)+(delu)/(dels)(dels)/(dely)=(delu)/(delr)(-1)/(y^2)+(delu)/(dels)(0)`
`(delu)/(delz)=(delu)/(delr)(delr)/(delz)+(delu)/(dels)(dels)/(delz)=(delu)/(delr)(0)+(delu)/(dels)((-1)/z^2)`
`therefore x^2(delu)/(delx)+y^2(delu)/(dely)+z^2(delu)/(delz)=(delu)/(delr)-(delu)/(dels)-(delu)/(delr)+(delu)/(dels)`
`x^2(delu)/(delx)+y^2(delu)/(dely)+z^2(delu)/(delz)=0`
Hence proved.
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