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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

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ConceptReview of Complex Numbers‐Algebra of Complex Number

#### Question

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)

#### 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

Is there an error in this question or solution?

#### APPEARS IN

Solution 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 Concept: Review of Complex Numbers‐Algebra of Complex Number.
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