What is the answer to solvefor x,y+e^y=(x+C)e^{-x} ?

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

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solvefor x,y+e^y=(x+C)e^{-x}

solve for

x, y + e^{y} = (x + C) e^{- x}

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

Solution steps

y + e^{y} = (x + C) e^{- x}

Rewrite the equation with

- x - C = u

and

x = - u - C

y + e^{y} = (- u - C + C) e^{- (- u - C)}

Simplify

y + e^{y} = - e^{u + C} u

Rewrite

y + e^{y} = - e^{u + C} u

in Lambert form:

e^{u} u = - \frac{y + e ^{y}}{e ^{C}}

Solve

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

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

Substitute back

u = - x - C,

solve for

x

Solve

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

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

2. 3^{x} = 3^{5} + 17. 3^{5}

6^{2 n + 1} + 6^{2} = 6^{n} + 6^{n + 3}

x^{l o g_{10} (4 x)} = 256

i, x^{i} = 2

9^{x} = 2 7^{x}

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