I have some problems in solving the following differential system.
To simplify notation, I write $x$ for $x(t)$ and $x'$ for $\frac{\text{d}}{\text{d} t} x(t)$
$$ \begin{cases} w'=w^{a+1} & \\ x'=x^{b+1}w^{a-b}& \\ y'=y^{c+1}w^{a-c}& \\ z'=z^{d+1}w^{a-d}& \\ \end{cases} $$
The initial conditions are
$$ \begin{cases} w(0)=w_0&\\ x(0)=x_0& \\ y(0)=y_0& \\ z(0)=z_0& \\ \end{cases} $$
Progress
Until now I have solved the equation in $w$ directly, getting $w(t)=w_0(1-aw_0^at)^{-1/a}$. But then I am not sure how to continue.
$$ w' = w^{a+1}\\ x^{-b-1}x' = \frac{-1}{b}\left(x^{-b}\right)' = w^{a-b}\\ y^{-c-1}y' = \frac{-1}{c}\left(y^{-c}\right)' =w^{a-c}\\ z^{-d-1}z' = \frac{-1}{d}\left(z^{-d}\right)' = w^{a-d} $$ also we can do $$ w^{-n -1}w' = \frac{-1}{n}\left(w^{-n}\right)' = w^{a-n} $$ where $n$ can be any of the $b,c$ or $d$. Does this help you.