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2. FIRST-ORDER DIFFERENTIAL EQUATIONS
Problem (Page 52 # 24a). A rock contains two radioactive isotopes, RA1
and RA2, that belong to the same radioactive series; that is, RA1 decays into
RA2, which then decays into stable atoms. Assume that the rate at which
RA1 decays into RA2 is 50e 10t kg/sec. Because the rate of decay of RA2 is
proportional to the mass y(t) of RA2 present, the rate of change in RA2 is
dy
= rate of creation rate of decay,
dt
dy
= 50e 10t ky,
dt
where k > 0 is the decays constant. If k = 20/sec and initially y(0) = 40 kg,
find the mass y(t) of RA2 for t > 0. The IVP in standard form is then
dy
+ 20y = 50e 10t, y(0) = 40.
dt
R
µ=e
Then
20 dt
= e20t.
e20ty 0 + 20e20ty = 50e10t =)
d h 20t ⇤
e y = 50e10t =)
dt
Z
e20ty = 50
e10t dt + C =)
⇣1⌘
20t
e y = 50
e10t + C =)
10
y(t) = 5e 10t + Ce 20t.
y(0) = 5 + C = 40 =) C = 35.
Thus
y(t) = 5e
10t
+ 35e
20t
.