# Solve $W_0(x)^2 - W_{-1}(x)^2 = c$ for $x<0$ with sufficiently large $c>0$

Suppose we have $$W_0(x)^2 - W_{-1}(x)^2 = c$$ for some constant $$c>0$$ with $$x<0$$. Then can we solve for $$x$$ algebraically? Or at least analytically find bounds for $$x$$?

Here we can assume $$c$$ is very large. The equation is from this link.

Also, we may represent $$x$$ in other forms; for example, $$x \leftarrow e^{-x-1}$$ changing the range of $$x$$ to $$0 and then use bounds.

Since $$W_0(x)^2<1$$, $$W_{-1}(x)^2>1$$ and hence $$W_0(x)^2-W_{-1}(x)^2<0$$ $$\forall x\in(-\tfrac1{\mathrm e},0)$$, there is no solution for $$c>0$$.