Up: [[공학수학1,engineering_mathematics_1]]
and [[수학,math]] [[모형,model]] = [[모델,model]]

== 개체군동태,population_dynamics ==

P(t): the total population at time t

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

== 방사성붕괴,radioactive_decay ==
A(t): amount of substance remaining at time t

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

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

== Newton's law of cooling/warming ==
뉴턴_냉각법칙

''T''(t): the temperature of a body at time t
''T'',,m,,: the temperature of surrounding medium

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

== 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)

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

== chemical reactions (이하 수업에서 다루지는 않음) ==
$\frac{dX}{dt}=k(\alpha-X)(\beta-X)$

== mixture ==
(input rate of salt) - (output rate of salt) = R,,in,, - R,,out,,
[[혼합물,mixture]]

== draining a tank ==
$\frac{dh}{dt}=-\frac{A_h}{A_w}\sqrt{2gh}$

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

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

== falling bodies and air resistance ==
$m\frac{dv}{dt}=mg-kv^2$

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

== a slipping chain ==
$\frac{d^2x}{dt^2}-\frac{64}Lx=0$

== suspended cables ==
$\frac{dy}{dx}=\frac{W}{T_1}$