3.1 Modeling with 1st Order Linear Differential Equations (202)
1. Law of Growth and Decay Model:
Rate of Change in a Substance is proportional to the Substance
dx
= kx; x(t0 ) = x0
dt
Example: The radioactive isotope of lead ‘Pb-209’ decays at a rate proportional to
the amount present at time t and has a half-life of 3.3 hours.
If 1 gram of the lead is present initially, how long will it take for 90% of
the lead to decay?
2. Newton’s Law of Cooling
Rate of Change in the Temperature of a body is proportional to the
Difference of the Temperature of the Object and Temperature of the
Surrounding
dT
= k (T − Ts ); T (t0 ) = T0
dt
Example: A small metal bar, whose initial temperature is 20oC is dropped
into a container of boiling water.
o
(a) How long will it take the bar to reach 90oC if its temperature increases 2
in 1 second?
(b)
How long will it take the bar to reach 98oC?
3. Mixture of Two Solutions Model
When two Brine (Salt Water) Solutions are mixed then the Rate of
Change in the Amount of Salt in the mixing Tank is proportional to the
Difference of the Input and Output Rate of salt
dA
= k ( Ri − Ro )
dt
Example: The A large tank contains 200 liters of fluid in which 30 gms of salt is
dissolve. Brine containing 1 gm of salt per liter is then pumped into the tank at the
rate of 4 liter/min.
Find the number A(t) of grams of salt in the tank at time t.
4. LRC-Circuits (Kirchoff’s 2nd Law)
Current in a Single Loop Circuit after switch is closed = i(t)
Current in a
Single Loop
Circuit after
switch is
closed = i(t)
Inductor
Inductance:= L
Current = i(t)
Voltage drop across: L
Capacitor
Capacitance:= C
Voltage drop across:
di
dt
Voltage drop across: R i(t)
Relationship between i and q:
i (t ) =
1
q (t )
C
LR-Circuit
Sum of Voltage drop across
inductor & Voltage drop across
resistor
=
Impressed Voltage on Circuit
L
Resistor
Resistance:= R
di
+ Ri = E (t )
dt
dq
dt
Impressed Voltage on Circuit := E(t)
RC-Circuit
Sum of Voltage drop across
resistor & Voltage drop across
capacitor
=
Impressed Voltage on Circuit
R
dq 1
+ q = E (t )
dt C
LRC-Circuit
Sum of Voltage drop across inductor & Voltage drop across resistor
=
Impressed Voltage on Circuit
L
1
di
+ Ri + q = E (t )
dt
C
Example: A 200-volt electromotive force is applied to an RC series circuit in which
the resistance is 1000 ohms and the capacitance is 5x10 -4 farad.
(a)
(b)
(c)
Find the charge q(t) on the capacitor if i(0) = 0
Determine the charge and the current at t = 0.005 s.
Determine the charge as t → ∞ .