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

The answer to solvefor x,c=e^x+xy-y^2 is x=-(\W(1/(e^{\frac{-c-y^2){y}}y})y-c-y^2)/y

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solvefor x,c=e^x+xy-y^2

solve for

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

x = - \frac{W ( \frac{1}{e ^{\frac{- c - y ^{2}}{y}} y} ) y - c - y ^{2}}{y}

Solution steps

c = e^{x} + x y - y^{2}

Prepare

c = e^{x} + x y - y^{2}

for Lambert form:

1 = (c - x y + y^{2}) e^{- x}

Rewrite the equation with

- x + \frac{c + y ^{2}}{y} = u

and

x = - \frac{y u - c - y ^{2}}{y}

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

Simplify

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

Rewrite

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

in Lambert form:

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

Solve

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

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

Substitute back

u = - x + \frac{c + y ^{2}}{y},

solve for

x

Solve

- x + \frac{c + y ^{2}}{y} = W (\frac{1}{e ^{\frac{- c - y ^{2}}{y}} y}) : x = - \frac{W ( \frac{1}{e ^{\frac{- c - y ^{2}}{y}} y} ) y - c - y ^{2}}{y}

x = - \frac{W ( \frac{1}{e ^{\frac{- c - y ^{2}}{y}} y} ) y - c - y ^{2}}{y}

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

3^{x + 1} + 3^{2 - x} = 28

3^{2 x} - 5 (3^{x}) - 36 = 0

83 = 1 5^{x}

50 = \frac{160}{1 + 7 e ^{- 0.3 t}}

What is the answer to solvefor x,c=e^x+xy-y^2 ?
The answer to solvefor x,c=e^x+xy-y^2 is x=-(\W(1/(e^{\frac{-c-y^2){y}}y})y-c-y^2)/y