What is the answer to solvefor x,xy=e^{2x+y} ?

The answer to solvefor x,xy=e^{2x+y} is x=-(\W(-(2e^y)/y))/(2)

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solvefor x,xy=e^{2x+y}

solve for

x, x y = e^{2 x + y}

x = - \frac{W ( - \frac{2 e ^{y}}{y} )}{2}

Solution steps

x y = e^{2 x + y}

Prepare

x y = e^{2 x + y}

for Lambert form:

x y e^{- 2 x - y} = 1

Rewrite the equation with

- 2 x = u

and

x = - \frac{u}{2}

(- \frac{u}{2}) y e^{- 2 (- \frac{u}{2}) - y} = 1

Simplify

- \frac{e ^{u - y} y u}{2} = 1

Rewrite

- \frac{e ^{u - y} y u}{2} = 1

in Lambert form:

e^{u} u = - \frac{2 e ^{y}}{y}

Solve

e^{u} u = - \frac{2 e ^{y}}{y} : u = W (- \frac{2 e ^{y}}{y})

u = W (- \frac{2 e ^{y}}{y})

Substitute back

u = - 2 x,

solve for

x

Solve

- 2 x = W (- \frac{2 e ^{y}}{y}) : x = - \frac{W ( - \frac{2 e ^{y}}{y} )}{2}

x = - \frac{W ( - \frac{2 e ^{y}}{y} )}{2}

6 4^{x - 1} = 1 6^{2 x - 2}

14 (2 + e^{- x})^{- 1} = 3

5 (5^{2 x}) - 26 (5^{x}) + 5 = 0

\frac{13}{2 + e ^{x}} = 4

7^{6 x + 7} = 3

What is the answer to solvefor x,xy=e^{2x+y} ?
The answer to solvefor x,xy=e^{2x+y} is x=-(\W(-(2e^y)/y))/(2)