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I have a differential equation which models a specific diet.

$\frac{dw}{dt} = -{w}^3 - {w}^2$ (leaving out constants for simplicity)

Wolfram Alpha tells me the solution is:

$log(w+1) - log(w) - \frac{1}{w} = c - t$

Is it possible to rearrange this to make w the subject of the equation?

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    Do you have any experience with differential equations? This one is separable. – 2017-01-08
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    Do you mean you want to solve the equation for $w$ in terms of $t$? I would say probably unlikely. WolframAlpha does not give a solution: https://www.wolframalpha.com/input/?i=solve+log(w%2B1)-log(w)-1%2Fw%3Dc-t+for+w – 2017-01-08
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    Yeah - it is separable. You separate it and then use partial fractions and I'm sure you'll get the same solution that wolfram alpha generated... – 2017-01-08
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    @Rahul - yes exactly. Ahhh I see, thats a bummer. – 2017-01-08

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$$\log(w+1) - \log(w) - \frac{1}{w} = c - t \\ \implies\log\left(\left(1+\frac{1}{w}\right)\exp\left(-\left(1+\frac{1}{w}\right)\right)\right)=c-1-t$$

Let $v=-\left(1+\frac{1}{w}\right)$, then

$$\implies -v\exp(v)=\exp(c-1-t) \\ \implies v\exp(v)=-\exp(c-1-t) \\ \implies\text{Lambert_W}(-\exp(c-1-t))=v \\ \implies w=\frac{-1}{1+v}=\frac{-1}{1+\text{Lambert_W}(-\exp(c-1-t))}$$