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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 $0<x$ and then use bounds.

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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 $c>0$.

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