Consider the following equation: $$y-\ln{y}=F(x,k) \ \ \ \ \ \ x,y,k \in \mathbb{R}$$ where $F$ is a non linear real function of $x$ and a constant $k$, that we know. Suppose that we want to use python, lets say with the standard module matplotlib, to plot $y$ as a function of $x$; to do this we would like to find an analytical expression for $y(x)$ (or maybe it's better to say a closed-form expression for $y(x)$, I am not really familiar with the terminology), problem is: there is no way to express $y(x)$ in a closed form.
The question now is: how can I plot the solutions to this equation?
My first instinct is to improvise, with python, some sort of search algorithm: we pick an arbitrary value for $x=x_0$, we find $F(x_0,k)$ and then we implement a search algorithm to find what value of $y$, once plugged into the left side, gives $F(x_0,k)$. We now have a point of the graph! We now interate the procedure, we find multiple points, and then connect the dots; and we get our plot. Does this procedure have a name? Is there a better, or more standard, way to make a plot of this? If my procedure is the better one: is there a standard search algorithm to use?