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 problems, 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 bacteria 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 function and one or more of its derivatives. Definition (ODE) Ordinary differential equations: differential equations that involve 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 function and one or more of its derivatives. Definition (ODE) Ordinary differential equations: differential equations that involve 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 function and one or more of its derivatives. Definition (ODE) Ordinary differential equations: differential equations that involve 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