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Error propagation in an equation without an analytical solution

Is it possible to formally propagate uncertainties through an equation which does not have an analytical solution? I am an Earth Scientist working on doing a rigorous determination of the age calculated using (U-Th)/He dating, which uses the equation

$$
He = 8*^{238}U(e^{lambda_{238}t}-1)times7*^{235}U(e^{lambda_{235}t}-1)times6*^{232}Th(e^{lambda_{232}t}-1)times^{147}Sm(e^{lambda_{147}t}-1)
$$

We solve for t numerically because the equation cannot be solved for t explicitly. The other variables (e.g., He, 238U, lambda238) are measured or known, each with their own uncertainty. For the sake of argument, let’s say that I know the uncertainty on each value perfectly and all are gaussian. Is there a way to determine the uncertainty in t using "classical error propagation" (e.g., adding in quadrature) without the ability to solve for it explicitly? I know I could build a monte carlo code to figure it out, but I’m wondering if it’s even possible to determine the uncertainty analytically.

Cross Validated Asked by slekce on November 14, 2021

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