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If U= F ( Y − X X Y , Z − X X Z ) , Show that X 2 ∂ U ∂ X + Y 2 ∂ U ∂ Y + X 2 ∂ U ∂ Z = 0 - Applied Mathematics 1

If u=`f((y-x)/(xy),(z-x)/(xz)),"show that"  x^2 (del_u)/(del_x)+y^2 (del_u)/(del_y)+x^2 del_u/del_z=0`

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Solution

Let  u=f(r,s)

∴` r=( y-x)/(xy)`                      ∴ `s=(z-x)/(xz)`

∴ `del_u/del_x=(del_u del_r)/(del_r del_x)+(del_u del_s)/(del_s del_x)=(del_u 1)/(del_r x^2)+del_u/del_s((-1)/x^2)` 

`del_u/del_y=(del_u del_r)/(del_r del_x)+(del_u del_s)/(del_s del_y)=del_(-1)/(del_r y^2)+del_u/del_s(0)` 

`del_u/del_z=(del_u del_r)/(del_r del_z)+(del_u del_s)/(del_s del_z)=del_u/del_r(0)+del_u/del_s(1/z^2)` 

∴ `x^2 del_u/del_x+y^2 del_u/del_y+z^2 del_u/del_z=del_u/del_r-del_u/del_s-del_u/del_r+del_u/del_s` 

`x^2 del_u/del_x+y^2 del_u/del_y+z^2 del_u/del_z=0`

Hence proved. 

Concept: .Circular Functions of Complex Number
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