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