Interesting first order ODEs in applied science I'm teaching first and second order ODEs this term and would like some nice examples that are easy enough to solve. I have plenty of second order ODEs, e.g. simple/damped/driven harmonic motion. As for first order ODEs, I have alreay used the following:


*

*$M' = - kM$ (nuclear half-life)

*$\theta' = -k(\theta-R)$ (Newton's law of cooling in a room of temperature $R$)

*$P' = kP(T-P)$ (Infection in a population of $T$)


Can anyone suggest any other first order ODEs which can be solved and that are used in applied sciences? Any help would be greatly appreciated.
 A: There about a googolplex of such examples, so I'll add a few to your list:
1.)  Voltage $V_C$ on a grounded capacitance $C$ driven by a voltage source $V(t)$ through a resistance $R$:  $RC \dot V_C + V_C = V(t)$;
2.)  Current $I_L$ through a grounded inductance $L$ driven by a voltage source $V(t)$ through a resistance $R$:  $L \dot I_L + RI_L = V(t)$;
Nota Bene:  In either example (1) or (2), the capacitance or inductance may be assumed to be non-linear functions $C(V_C)$, $L(I_L)$ of the voltage $V_C$ or current $I_L$, respectively, thus providing examples of non-linear, first order systems.  The resistance $R$ may also be similarly non-linearized in various ways; for example, a semiconductor diode is often assumed to exhibit an exponential voltage-current relationship of the form $I_D = I_S e^{\alpha V_D}$, where $I_D$ is the current at applied voltage $V_D$, and $I_S$, $\alpha$ are constant related to the physics and construction of the device, so it is effectively modelled as a type of non-linear resistance.  See the Wikipedia article http://en.m.wikipedia.org/wiki/Diode for more information including further references.  Finally,  equations of the generic type (1), (2) may be turned into second-order systems via the addition of an inductance to (1) or a capacitance to (2); this provides a convenient conceptual linkage 'twixt first and second order equations which may be useful for pedagological purposes.  Basic electronic circuit theory is rife with such instances, and the perusal of any of a number of standard texts should provide many more helpful examples.  End of Note.
3.)  An inertial particle of mass $m$ moving only under the influence of linear damping forces and no others, such as elastic forces, may be modelled by an equation of the form $m \ddot x + c \dot x = 0$, where $c$ is a constant describing the strength of the frictive forces.  The linear term $c \dot x$  may be replaced by a more general, non-linear term such as $c (\dot x)^\beta$, $\beta$ a system-dependent constant, in certain situations, leading to $m \ddot x + c (\dot x)^\beta = 0$.
4.)  Check out the rocket equation here: http://en.m.wikipedia.org/wiki/Tsiolkovsky_rocket_equation.
5.)  Last but by no means least, the grand-daddy of them all, the equation for exponential growth of a colony of bacteria (or almost anything else), $\dot P = kP$, $P(t)$ being the population at tome $t$ and $k$ the growth constant.  
There are oh-so-many more examples but work beckons and so I must leave you with these.
Hope this helps.  Cheerio,
and as always,
Fiat Lux!!!
A: 1) Fall of a parachute with viscous (linear) friction or realistic (quadratic) friction.
$$\ddot y=-g-k\dot y$$ vs. $$\ddot y=-g-k\dot y^2.$$
2) trajectory of a cannonball, with viscous friction.
