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If Z = Log ( E X + E Y ) Show that Rt − S 2 = 0 Where R = ∂ 2 Z ∂ X 2 , T = ∂ 2 Z ∂ Y 2 S = ∂ 2 Z ∂ X ∂ Y - Applied Mathematics 1

If `z =log(e^x+e^y) "show that rt" - s^2 = 0  "where r"= (del^2z)/(delx^2),t=(del^2z)/(dely^2)"s"=(del^2z)/(delx dely)`

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Solution 1

`z =log(e^x+e^y)`

(1)`(delz)/(delx)=e^x/(e^x+e^y)`

`(del^2z)/(delx^2)=(e^x(e^x+e^y)-e^x(e^x))/(e^x+e^y)^2=(e^(2x)+e^(xy)-e^(2x))/(e^x+e^y)^2`

`r=(del^2z)/(delx^2)=e^(xy)/(e^x+e^y)^2`  .............(1)

(2) `(delz)/(dely)=e^y/(e^x+e^y)`

`(del^2z)/(dely^2)=(e^y(e^x+e^y)-e^y(e^y))/(e^x+e^y)^2=(e^(2y)+e^(xy)-e^(2y))/(e^2+e^y)^2`

t `=(del^2z)/(dely^2)=e^(xy)/(e^x+e^y)2`   ................(2)

Solution 2

`z =log(e^x+e^y)`

(1)`(delz)/(delx)=e^x/(e^x+e^y)`

`(del^2z)/(delx^2)=(e^x(e^x+e^y)-e^x(e^x))/(e^x+e^y)^2=(e^(2x)+e^(xy)-e^(2x))/(e^x+e^y)^2`

`r=(del^2z)/(delx^2)=e^(xy)/(e^x+e^y)^2`  .............(1)

(2) `(delz)/(dely)=e^y/(e^x+e^y)`

`(del^2z)/(dely^2)=(e^y(e^x+e^y)-e^y(e^y))/(e^x+e^y)^2=(e^(2y)+e^(xy)-e^(2y))/(e^2+e^y)^2`

t `=(del^2z)/(dely^2)=e^(xy)/(e^x+e^y)2`   ................(2)

(3) `(delz)/(delx)=e^x/(e^x+e^y)^`

s `=(del^2z)/(delxdely)=e^(xy)/(e^x+e^y)^2` ................(3)

From (1), (2) and (3) we get,

rt `=(e^(xy)/(e^x+e^y)^2)xx(e^(xy)/(e^x+e^y)^2)=(e^(xy)/(e^x+e^y)^2)^2=(e^(2xy)/(e^x+e^y)^2)`  .............(4)

s2 = `(e^(xy)/(e^x+e^y)^2)^2= (e^(2xy)/(e^x+e^y)^2)`................(5)

From (4) and (5) we get,

rt-s2 = `(e^(2xy)/(e^x+e^y)^2)-(e^(2xy)/(e^x+e^y)^2)=0`

Hence proved rt - `s^2`= 0

Solution 3

`z =log(e^x+e^y)`

(1)`(delz)/(delx)=e^x/(e^x+e^y)`

`(del^2z)/(delx^2)=(e^x(e^x+e^y)-e^x(e^x))/(e^x+e^y)^2=(e^(2x)+e^(xy)-e^(2x))/(e^x+e^y)^2`

`r=(del^2z)/(delx^2)=e^(xy)/(e^x+e^y)^2`  .............(1)

(2) `(delz)/(dely)=e^y/(e^x+e^y)`

`(del^2z)/(dely^2)=(e^y(e^x+e^y)-e^y(e^y))/(e^x+e^y)^2=(e^(2y)+e^(xy)-e^(2y))/(e^2+e^y)^2`

t `=(del^2z)/(dely^2)=e^(xy)/(e^x+e^y)2`   ................(2)

(3) `(delz)/(delx)=e^x/(e^x+e^y)^`

s `=(del^2z)/(delxdely)=e^(xy)/(e^x+e^y)^2` ................(3)

From (1), (2) and (3) we get,

rt `=(e^(xy)/(e^x+e^y)^2)xx(e^(xy)/(e^x+e^y)^2)=(e^(xy)/(e^x+e^y)^2)^2=(e^(2xy)/(e^x+e^y)^2)`  .............(4)

s2 = `(e^(xy)/(e^x+e^y)^2)^2= (e^(2xy)/(e^x+e^y)^2)`................(5)

From (4) and (5) we get,

rt-s2 = `(e^(2xy)/(e^x+e^y)^2)-(e^(2xy)/(e^x+e^y)^2)=0`

Hence proved rt - `s^2`= 0

Concept: Review of Complex Numbers‐Algebra of Complex Number
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