수학적모델,mathematical_model


개체군동태,population_dynamics


P(t): the total population at time t

$\displaystyle \frac{dP}{dt}\propto P$

방사성붕괴,radioactive_decay

A(t): amount of substance remaining at time t

$\displaystyle \frac{dA}{dt}\propto A$

물질의 연대 측정법:
방사성 물질의 시간 t에서의 붕괴율(dy/dt)은 시간 t에서의 현존량(y)에 비례
$\displaystyle \frac{dy}{dt}=ky$

ex. 현재 Ra 100 mg 이 있다. t시간 이후 양은?
위 식을 변형하면
$\displaystyle \frac{dy}{ky}=dt$
$\displaystyle \frac{dy}{y}=kdt$
$\displaystyle \ln|y|=kt+C_1$
$\displaystyle y=e^{kt+C_1}=C_2e^{kt}$
$\displaystyle t=0,y=100,y=100e^{kt}$

Newton's law of cooling/warming

뉴턴_냉각법칙

T(t): the temperature of a body at time t
Tm: the temperature of surrounding medium

$\displaystyle \frac{dT}{dt}\propto T-T_m$
or
$\displaystyle {dT\over dt}=k(T-T_m)$

spreading of disease

x(t): the number of people who have contacted the disease at time t
y(t): the number of people who have not yet been exposed to the disease at time t
assumption: the number of interactions is jointly proportional to x(t) and y(t)

$\displaystyle \frac{dx}{dt}\propto xy$
or
$\displaystyle \frac{dx}{dt}=kxy$

chemical reactions (이하 수업에서 다루지는 않음)

$\displaystyle \frac{dX}{dt}=k(\alpha-X)(\beta-X)$

mixture

(input rate of salt) - (output rate of salt) = Rin - Rout
혼합물,mixture

draining a tank

$\displaystyle \frac{dh}{dt}=-\frac{A_h}{A_w}\sqrt{2gh}$

series circuit

$\displaystyle L\frac{d^2q}{dt^2}+R\frac{dq}{dt}+\frac1Cq=E(t)$

falling bodies

$\displaystyle \frac{d^2s}{dt^2}=-g$

falling bodies and air resistance

$\displaystyle m\frac{dv}{dt}=mg-kv^2$

$\displaystyle m\frac{d^2s}{dt^2}+k\left(\frac{ds}{dt}\right)^2=mg$

a slipping chain

$\displaystyle \frac{d^2x}{dt^2}-\frac{64}Lx=0$

suspended cables

$\displaystyle \frac{dy}{dx}=\frac{W}{T_1}$