# Failure rate of a system at time 't'

by francisg3
Tags: failure, rate, time
 P: 610 for the case 1 $$\lambda = \lambda_o (1-\frac{\mu}{\alpha})$$ $$\frac{d\mu}{dt}=\lambda_o(1-\frac{\mu}{\alpha})$$ $$\frac{d\mu}{\left(1-\frac{\mu}{\alpha}\right)}=\lambda_o \,\, dt$$ now integrate within the given limits
 P: 2 I am also having problems with this question, I integrated: $\int\frac{dμ}{(1-\frac{μ}{α})}$ = $\int$$\lambda$0dt And I got: -αln(μ-α) = $\lambda$0t I'm not sure if this is going in the right direction and what would I have to do after this in order to find μ(t) and λ(t)? Thanks
 P: 610 kazo, use properties of logarithm... $$-\alpha \ln (\mu-\alpha)=\lambda_o t$$ $$\ln (\mu-\alpha)=-\frac{\lambda_o t}{\alpha}$$ $$\mu-\alpha= \mbox{exp}\left[-\frac{\lambda_o t}{\alpha}\right ]$$ $$\mu (t) =\alpha +\mbox{exp}\left[-\frac{\lambda_o t}{\alpha}\right ]$$ and plug this to get $\lambda$ as function of t