同济大学：《常微分方程》课程教学资源（讲义）微分方程基本概念及几类可求解析解的方程

Ordinary Differential Equations Peipei Shang School of Mathematical Sciences shang@tongji.edu.cn 4口14①y4至2000 Peipei Shang School of Mathematical Sciences shang@tongji.edu.Ordinary Differential Equations

Ordinary Differential Equations Peipei Shang School of Mathematical Sciences shang@tongji.edu.cn Peipei Shang School of Mathematical Sciences shang@tongji.edu.cn Ordinary Differential Equations

Why do we study 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0y￥至无3000 Peipei Shang School of Mathematical Sciences shang@tongji.edu.Ordinary Differential Equations

Why do we study 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. Peipei Shang School of Mathematical Sciences shang@tongji.edu.cn Ordinary Differential Equations

Example Newton's law of cooling may be stated in this way:The time rate of change (the rate of change with respect to t)of the temperature T(t)of a body is proportional to the difference between T and the temperature A of the surrounding medium. That is dT =-k(T-A), dr where k is a positive constant.Observe that If T>A,then dT/dt0,T is increasing 4口14①y至，元2000 Peipei Shang School of Mathematical Sciences shang@tongji.edu.Ordinary Differential Equations

Example Newton’s law of cooling may be stated in this way: The time rate of change (the rate of change with respect to t) of the temperature T(t) of a body is proportional to the difference between T and the temperature A of the surrounding medium. That is dT dt = −k(T −A), where k is a positive constant. Observe that If T > A, then dT/dt 0, T is increasing Peipei Shang School of Mathematical Sciences shang@tongji.edu.cn 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) where k is an unknown constant.Note that P(t)=Ce,C0 is a solution of(1),because P'(t)=Cke=k(Ce)=kP(t);VIER 4口14①y至元2000 Peipei Shang School of Mathematical Sciences shang@tongji.edu.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) where k is an unknown constant. Note that P(t) = Ce kt , C > 0 is a solution of (1), because P 0 (t) = Ckekt = k(Cekt) = kP(t), ∀t ∈ R Q: What can we do with the solution? Peipei Shang School of Mathematical Sciences shang@tongji.edu.cn 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) where k is an unknown constant.Note that P(t)=Ce,C0 is a solution of (1),because P'(t)=Ckek =k(Ce)=kP(t);VtER Q:What can we do with the solution? 4口10y至，无2000 Peipei Shang School of Mathematical Sciences shang@tongji.edu.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) where k is an unknown constant. Note that P(t) = Ce kt , C > 0 is a solution of (1), because P 0 (t) = Ckekt = k(Cekt) = kP(t), ∀t ∈ R Q: What can we do with the solution? Peipei Shang School of Mathematical Sciences shang@tongji.edu.cn Ordinary Differential Equations

Example (To predict) Suppose that P(t)=Cekr is the population of a colony of bac- teria at time t(hours,h), 1000=P(o)=Ce0=C; C=1000: 12000=P(1)=Ce 1k=ln2≈0.693147 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14①y至元2000 Peipei Shang School of Mathematical Sciences shang@tongji.edu.Ordinary Differential Equations

Example (To predict) 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 = ln2 ≈ 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 Peipei Shang School of Mathematical Sciences shang@tongji.edu.cn Ordinary Differential Equations

Terminology Definition(DE) Differential equations:equations containing an unknown func- tion and one or more of its derivatives. 4口10y至，1元3000 Peipei Shang School of Mathematical Sciences shang tongji.edu.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 Peipei Shang School of Mathematical Sciences shang@tongji.edu.cn 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1日，4元卡2000 Peipei Shang School of Mathematical Sciences shang tongji.edu.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 Peipei Shang School of Mathematical Sciences shang@tongji.edu.cn 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 Peipei Shang School of Mathematical Sciences shang@tongji.edu.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 Peipei Shang School of Mathematical Sciences shang@tongji.edu.cn Ordinary Differential Equations

Example (Heat equation) The temperature u=u(x,t)of a long thin uniform rod satisfies dw,∂2u where k is the thermal diffusivity of the rod. 4口10y4至，元2000 Peipei Shang School of Mathematical Sciences shang tongji.edu.Ordinary Differential Equations

Example (Heat equation) 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. Peipei Shang School of Mathematical Sciences shang@tongji.edu.cn Ordinary Differential Equations