I was trying to find some examples of explicitly solving implicit functions. However, most I found was about implicit differentiation. For example, if we have a function $u(x,t)$, the implicit form says $$ u=\frac{1}{1+ut} $$
How can we directly invert this and got the explicit form for $u(x,t)$?
For a fixed time $t$, we can write the equation as
$$u(x)(1 + u(x)*t) = 1$$
$$\implies u + u^2t = 1$$
This is a quadratic in $u$, solutions given by
$$u(x) = \frac{-1 \pm \sqrt{1+4t}}{2t}$$
Now if we know $x(t)$ we can get the desired parametric form - usually this sort of problem is to find a quantity along a particular curve where we would have more information on the relationship of $x$ and $t$