When I opened up the "High Energy,Nuclear,Particle Physics" section, it was kind of like "Holy $#^*!!!- I don't know what any of the words mean!" I'm here because I don't know what I'm talking about. Also, I'm not sure if this is the right area of the site for this particular question. So, I had this idea of examining the radiation from an isotope with at least one daughter product decaying through a different mode than the parent, and then,possibly, using the ratio of mode to mode to estimate the composition and time of synthesis of the sample. It is the kind of thing where it seems as if it should be possible, but I'm just not sure of what type of math to throw at it. The amount of the parent product that exists is simply divided by two every half-life,but the whole concept of "half-life" almost just doesn't work to determine the amount of a daughter substance,as new material is constantly being added(from the decay of the parent, of course). I found it very difficult to exactly describe the problem I'm trying to solve, and ,as such, I apologize if you are left hopelessly confused by my thread.All replies are welcome! Also, I know it's kind of long...
In fact I don't understand the question... You want to examine the radiation from an isotope (let's say X) which decays to a daughter product (Y) [itex] X \rightarrow Y + Z [/itex] with Z the radiation (could be alpha or beta decay). then I don't understand how you continue.... like Y decaying by some other decay (not Z) to another element W? Well in that case the two modes are known... if the first is alpha decay the 2nd is beta or vice versa. parent product=? as many parents you have decaying, you get as many daughters. So if at some time you have the decay of ΔN parents, you have the creation of ΔN daughters (or in the equation of [itex]dN/dt|_{parents}[/itex] you should change the sign to get the daughters). What is the new material you have in mind? In general you don't have 1 equation... you should have 2 differential equations which are actually mixed. For example let's say you have: [itex] X \rightarrow Y [/itex] [itex] Y \rightarrow W [/itex] Then you should have that the number by which [itex]Y[/itex] changes by time is: [itex] \dot{N}_{Y} = n_{X} - n_{W} [/itex] In general you should also have some factors in front showing which is more favorable to happen, but in general I want to pass the idea of how you say that... So you have a + coming from the number of X decayed, which creates you Y particles... and you have - coming from the fact that your Y particles also decay to nW.
You do not need a nuclear reaction to be able to identify the composition of an unknown nucleus. All you need to do is shine a continuous light source on a sample and take some spectra of the scattered/emitted light from the sample. You can then analyse the resulting absorption lines in the spectra which are unique to that particular element/atom/molecule and then you can either compare the spectra with existing lab experiments with known substances or with computer models of excited states. It is possible in principle to analyse the radiation direct from a nuclear reaction/decay. Generally this sort of thing requires a comparison with radiation from a known sample in a lab though. However, this kind of analysis is done in order to identify elements in outer space. Really nice examples of this are Al-26 and Fe-60, which have isomeric states with very long half-lives and can therefore be used to 'carbon-date' stars and other celestial phenomena.
For a good summary of the math behind radioactive decay, try looking at this paper. I think what you are talking about is a decay chain, n_{1} → n_{2} → n_{3}. If so, scroll down to the section "Compound Decay".
If you are sure your initial sample is free of the decay product, you can simply put the probe in a mass spectrometer and measure the fraction of atoms that decayed. No decay chain needed, and most dating methods use this concept. If you cannot do this (because the sample is far away in space), the relative intensity between two decays in a chain could be used - but particles won't reach earth with the necessary energy resolution, so you need multiple gamma transitions. I'm not sure if that gives reliable results. See Bill_K's reference for formulas.