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

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

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

solve for

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

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

Solution steps

x y = e^{x + y}

Prepare

x y = e^{x + y}

for Lambert form:

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

Rewrite the equation with

- y = u

and

y = - u

x (- u) e^{- x - (- u)} = 1

Simplify

- e^{- x + u} xu = 1

Rewrite

- e^{- x + u} xu = 1

in Lambert form:

e^{u} u = - \frac{e ^{x}}{x}

Solve

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

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

Substitute back

u = - y,

solve for

y

Solve

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

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

9^{x^{2}} = 3^{x + 3}

3^{2 x} = 7

2^{3 x} - 2^{2 x} = 2^{x + 1}

9^{2 x - 1} = 3^{6 x}

12 5^{x} + 5^{3 x + 1} = 200

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