$y = 3 + x + e^{x}$ $y=3+x+e^x$ ....find the inverse of y?

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Answer 1 · 2018-04-15T14:50:26

$x = y - 3 - W (e^{y - 3})$ $x = y-3 - W(e^(y-3))$

Calling

$z = y - 3 - x$ $z = y-3-x$ we have

$z = e^{y - 3 - z} \Rightarrow z e^{z} = e^{y - 3} \Rightarrow z = W (e^{y - 3})$ $z = e^(y-3-z) rArr z e^z = e^(y-3) rArr z = W(e^(y-3))$

and finally

$x = y - 3 - W (e^{y - 3})$ $x = y-3 - W(e^(y-3))$

Here $W (\cdot)$ $W(cdot)$ is the Lambert function

y=3+x+e^xy=3+x+exy=3+x+e^x ....find the inverse of y?