Math 225 Differential Equations Notes Chapter 2

Example 4: Word Problem
A rock contains radioactive isotopes RA 1 and RA 2 that belong to
the same radioactive series:
• RA 1 decays into RA 2 at the rate of 50e −10t kg/sec.
• RA 2 decays into stable atoms at a rate proportional k to the
mass y(t) of RA 2 present
From this information we can setup a first order differential equation
describing the rate of change of RA 2 {y(t)}
dy
dt =
rate of creation - rate of decay
So
dy
dt = 50e−10t − ky(t)
If k = 2/sec and initially y(0) = 40 kg, find the mass y(t) of RA 2
for t ≥ 0
• dy
dt + 2y = 50e−10t Put in standard form so P (t) = 2
• ∫ P (t)dt = ∫ 2dx = 2t find µ(t)
• µ(t) = e 2t
• e 2t dy
dt + 2e2t y = 50e 2t e −10t Multiply the standard form by µ(t)
• d dt (e2t y) = 50e −8t
• e 2t y = ∫ 50e −8t dt Integrate both sides
• e 2t y = 50
−8 e−8t + C
• y = −25
4 e−10t + Ce −2t 10