Because of an unhealthy interest in mathematical problems I’ve been trying to figure out an equation for the amount of a radioactive product of radioactive decay.

The equation for the amount left after primary decay is of course:

x(t) = x0 * e^-Lt

where x0 is the initial amount and L is the decay constant. But if the result of this decay is a radioactive isotope y with decay constant l, what is the equation for the amount of y at time t?

I think it should be a solution to the differential equation:

y’ = -ly + x0 * L * e^-Lt

but I’m not sure I can solve that, and I’m not sure I haven’t made any wrong assumptions, and I should have spent the last two hours doing something else besides playing around with this. So any knowledgable input would be very welcome, or I might waste most of tomorrow’s workday as well.