What is the answer to solvefor x,xy+e^{xy}=0 ?

The answer to solvefor x,xy+e^{xy}=0 is x=-(\W_{0}(1))/y

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solvefor x,xy+e^{xy}=0

solve for

x, x y + e^{x y} = 0

x = - \frac{W _{0} ( 1 )}{y}

Solution steps

x y + e^{x y} = 0

Prepare

x y + e^{x y} = 0

for Lambert form:

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

Rewrite the equation with

- x y = u

and

x = - \frac{u}{y}

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

Rewrite

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

in Lambert form:

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

Solve

- e^{u} (- \frac{u}{y}) y = 1 : u = W_{0} (1)

u = W_{0} (1)

Substitute back

u = - x y,

solve for

x

Solve

- x y = W_{0} (1) : x = - \frac{W _{0} ( 1 )}{y}

x = - \frac{W _{0} ( 1 )}{y}

9^{x} = \frac{1}{3}

e^{- x^{2}} = e^{- 3 x - 4}

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

2^{x} = \frac{3}{2}

4^{20} + 4^{20} = 2^{x}

What is the answer to solvefor x,xy+e^{xy}=0 ?
The answer to solvefor x,xy+e^{xy}=0 is x=-(\W_{0}(1))/y