Posted by **Quinn** on Thursday, January 17, 2013 at 10:08pm.

Model radioactive decay using the notation

t = time (independent variable)

r(t) = amount of particular radioactive isotope present at time t (dependent variable)

-λ = decay rate (parameter)

Note that the minus sign is used so that λ > 0

a) Using this notation, write a model for the decay of a particular radioactive isotope.

b) If the amount of the isotope present at t = 0 is r0, state the corresponding initial-value problem for the model in part (a).

- Differential Equations -
**bobpursley**, Thursday, January 17, 2013 at 10:27pm
the rate is proportional to r(t) times a constant

dr/dt=r(t)* constant where the constant is negative

and the solution to this first order diff equation is of the form

r(t)=K*e^(- λ)t + C

b. ro=K+C

and at t=inf, r(inf)=0

which implies C is zero, so k=ro.

r(t)=ro*e^- λ t

- Differential Equations -
**Ryan Ward**, Thursday, February 7, 2013 at 12:04am
a) dr(t)/dt = -λr(t)

=>∫[ dr(t)/r(t) ] = ∫[ -λt ]

=> ln |r| = -λt + C

=> r(t) = e^(-λt + C) = e ^ (C - λt)

=> r(t) = e^C * e ^ (-λt)

let C = e^C

=> r(t) = Ce^(-λt)

b) r(0) = r_0

=>r(0) = r_0 = Ce^(-λ(0))

=>r_0 = C

=>r(t) = r_0 * e^(-λt)

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