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DiffEq Cheat Sheet (DRAFT) by

DiffEq

This is a draft cheat sheet. It is a work in progress and is not finished yet.

3.1 Modeling with Linear DE

Radioa­ctive Decay
x(t)=x­_0*­e^(kt)
Population
P(t) = P_0*e^(kt)
Compound Interest
S(t)=S­_0*­e^(rt)
Coolin­g/H­eating
T(t)=T­_m+­(T_­0-T­_m)­*e^(kt)
Mixing of Two Fluids
dA/dt=­R_i­n-R_out
 

Examples

Population increases propor­tional to #people at time t. If initial population P0 has doubled in 7 years, how long will it take to triple­/qu­adr­uple?
2P0/P0­=e^(kt) => ln(2)=kt => k=ln(2)/7 => (ln(2)/7)
t(trip­)=l­n(3­)/[­ln(­2)/7] = 11.09yr => t(quad­)=l­n(4­)/[­ln(­2)/7] = 14yr

A tank contains 1000 L pure h2o. Brine w/ 1 kg/L salt pumped in @ 6L/min; the well-mixed solution is pumped out at the same rate. When will concen­tration = 0.5kg/L
A'(t)=­Rin­-Rout => Rin=6k­g/min; Rout=(­A/1­000­kg/­L)*­(6L­/mi­n)=­3A/500 kg/min
A'(t)+­3A/­500=6 => A(t)=1­000­+Ce­^(-­3t/500) => A(0)=0 => C=-1000
1000=1000e(-3t/5­00)=500 => e(-3t/5­00)=1/2 => t=115.53s