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/dt<0,T is decreasing If T<A,then dT/dt>0,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 decreasing 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