同济大学：《常微分方程》课程教学资源（讲义）First-order differential equations

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 prob￾lems, 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 bac￾teria 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 bac￾teria 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

Mathematical Models Real-world situation Formulation interpretation Mathematical Model Mathematical Analysis Mathematical results Figure:The process of mathematical modeling Ordinary Differential Equations

Mathematical Models Real-world situation Formulation Interpretation Mathematical Model Mathematical Analysis Mathematical results Figure: The process of mathematical modeling Ordinary Differential Equations

Terminology Definition(DE) Differential equations:equations containing an unknown func- tion and one or more of its derivatives. 4日10y至，1元3000 Ordinary Differential Equations

Terminology Definition (DE) Differential equations: equations containing an unknown func￾tion and one or more of its derivatives. Definition (ODE) Ordinary differential equations: differential equations that in￾volve an unknown function of a single independent variable . Example The population function P(t) dP dt = kP Ordinary Differential Equations

Terminology Definition(DE） Differential equations:equations containing an unknown func- tion and one or more of its derivatives. Definition (ODE) Ordinary differential equations:differential equations that in- volve an unknown function of a single independent variable. 4日10,1元2000 Ordinary Differential Equations

Terminology Definition (DE) Differential equations: equations containing an unknown func￾tion and one or more of its derivatives. Definition (ODE) Ordinary differential equations: differential equations that in￾volve an unknown function of a single independent variable . Example The population function P(t) dP dt = kP Ordinary Differential Equations

Terminology Definition (DE) Differential equations:equations containing an unknown func- tion and one or more of its derivatives. Definition (ODE) Ordinary differential equations:differential equations that in- volve an unknown function of a single independent variable. Example The population function P(t) dP =kP dr 4口14①y至，元2000 Ordinary Differential Equations

Terminology Definition (DE) Differential equations: equations containing an unknown func￾tion and one or more of its derivatives. Definition (ODE) Ordinary differential equations: differential equations that in￾volve an unknown function of a single independent variable . Example The population function P(t) dP dt = kP Ordinary Differential Equations

Example The temperature u=u(x,t)of a long thin uniform rod satisfies du 2u 01-kx2 where k is the thermal diffusivity of the rod. 4日10y4至，1无2000 Ordinary Differential Equations

Example The temperature u = u(x,t) of a long thin uniform rod satisfies ∂u ∂ t = k ∂ 2u ∂ x 2 , where k is the thermal diffusivity of the rod. Definition (PDE) Partial differential equations: differential equations that involve an unknown function of more than one independent variables, together with partial derivatives of the function. Ordinary Differential Equations