Ordinary Differential Equations 4口14①y4至2000 Ordinary Differential Equations
Ordinary Differential Equations Ordinary Differential Equations
The laws of the universe are written in the language of mathematics.Algebra is sufficient to solve many static prob- lems,but the most interesting natural phenomena involve change and are described by equations that relate changing quantities. 4口1日，1元2000 Ordinary Differential Equations
The laws of the universe are written in the language of mathematics. Algebra is sufficient to solve many static problems, but the most interesting natural phenomena involve change and are described by equations that relate changing quantities. Ordinary Differential Equations
Example The time rate of change of a population P(t)with constant birth and death rates is proportional to the size of the population,i.e., dp (1) Note that P(t)=Cek is a solution of(1),because P'(t)=Cke=k(Ce)=kP(t),VIER 4日10y4至，1无2000 Ordinary Differential Equations
Example The time rate of change of a population P(t) with constant birth and death rates is proportional to the size of the population, i.e., dP dt = kP. (1) Note that P(t) = Ce kt is a solution of (1), because P 0 (t) = Ckekt = k(Cekt) = kP(t), ∀t ∈ R Ordinary Differential Equations
Example Suppose that P(r)=Cekt is the population of a colony of bac- teria at time t(hours,h), {om8二c2-c- c=1000, 2000=P(1)=Ck 1k=ln2≈0.693147 Thus, P(t)=1000.2 4口14①y4至2000 Ordinary Differential Equations
Example Suppose that P(t) = Cekt is the population of a colony of bacteria at time t (hours, h), ( 1000 = P(0) = Ce0 = C, 2000 = P(1) = Cek =⇒ ( C = 1000, k = ln 2 ≈ 0.693147 Thus, P(t) = 1000 · 2 t To predict the number of bacteria in the population after one and a half hours (t=1.5) is P(1.5) = 1000 · 2 3 2 ≈ 2828 Ordinary Differential Equations
Example Suppose that P(t)=Cekt is the population of a colony of bac- teria at time t(hours,h), {=8-c一-0m 2000=P(1)=Cek Thus, P(t)=1000.2 To predict the number of bacteria in the population after one and a half hours(t=1.5)is P(1.5)=1000.22≈2828 4口14①y4至2000 Ordinary Differential Equations
Example Suppose that P(t) = Cekt is the population of a colony of bacteria at time t (hours, h), ( 1000 = P(0) = Ce0 = C, 2000 = P(1) = Cek =⇒ ( C = 1000, k = ln 2 ≈ 0.693147 Thus, P(t) = 1000 · 2 t To predict the number of bacteria in the population after one and a half hours (t=1.5) is P(1.5) = 1000 · 2 3 2 ≈ 2828 Ordinary Differential Equations