[jacobcolbert]

I recently have been trying to model 2 step nuclear decay. I said that X decays into Y with a rate constant k_x. And Y decays into Z with rate constant k_x. I then set up the differential equation dy/dt=k_xx-k_yY. I then substituted in X=X₀e^-k_xt for X in the differential equation. The result when I solve the equation however has the term (k_y-k_x) in the demominator of a fraction which makes no sense to me because because what if they are equal? And what if k_x is greater than k_y? There shouldnt be a negative amount of Y. Once i find a place to host my work as a picture I can post that. Anyways if anyone knows of a correct equation to model this or could confirm or reject my original differential equation it would be appreciated. Thanks.

[Mitch]

The equation doesn't work when ky equals kx. Great catch! Are you sure you are in high school though?

[Yggdrasil]

In the case where k_x = k_y = k you can derive a different model for Y by choosing a particular solution of the form te^kt.

In the case where k_x > k_y the equation will still be positive since this implies that e^k_xt < e^k_yt for all t > 0 and two negative expressions will multiply to give a positive value.