Terminology Definition (Order) The order of an ODE is the highest order derivative that appears in the equation. Example (1)y+2xy=0;first order) (2)xy+y=x2;second order) (3)y/-x(y)3+y=0;third order) 4日10y4至，1无2000 Peipei Shang School of Mathematical Sciences shang tongji.edu.Ordinary Differential Equations
Terminology Definition (Order) The order of an ODE is the highest order derivative that appears in the equation. Example (1) y 0 +2xy = 0; ( first order) (2) xy00 +y 0 = x 2 ; ( second order) (3) y 000 −x(y 0 ) 3 +y = 0; ( third order) (4) d 2 y dt 2 +y = cos 2t; (second order) Peipei Shang School of Mathematical Sciences shang@tongji.edu.cn Ordinary Differential Equations
Terminology Definition (Order) The order of an ODE is the highest order derivative that appears in the equation. Example (1)y+2xy=0;first order) (2)xy+y=x2;second order) (3)y-x(y)3 +y=0;third order) (4)9 +y=c0s2r 4日10y4至，1无2000 Peipei Shang School of Mathematical Sciences shang tongji.edu.Ordinary Differential Equations
Terminology Definition (Order) The order of an ODE is the highest order derivative that appears in the equation. Example (1) y 0 +2xy = 0; ( first order) (2) xy00 +y 0 = x 2 ; ( second order) (3) y 000 −x(y 0 ) 3 +y = 0; ( third order) (4) d 2 y dt2 +y = cos 2t; (second order) Peipei Shang School of Mathematical Sciences shang@tongji.edu.cn Ordinary Differential Equations
Terminology Definition (Order) The order of an ODE is the highest order derivative that appears in the equation. Example (1)y+2xy=0;first order) (2)xy+y=x2;second order) (3)yx(y)3+y=0;(third order) (4)9 +y=cos2r (second order) 4口14①y至元2000 Peipei Shang School of Mathematical Sciences shang@tongji.edu.Ordinary Differential Equations
Terminology Definition (Order) The order of an ODE is the highest order derivative that appears in the equation. Example (1) y 0 +2xy = 0; ( first order) (2) xy00 +y 0 = x 2 ; ( second order) (3) y 000 −x(y 0 ) 3 +y = 0; ( third order) (4) d 2 y dt2 +y = cos 2t; (second order) Peipei Shang School of Mathematical Sciences shang@tongji.edu.cn Ordinary Differential Equations
Remark A nth-order ODE (An ODE of order n)can always be written in the general form Fy,y,…,y)=0 (2) here F is a given function of n+2 variables;y is the unknown function;x is the independent variable.(2)is called implicit equation. 4口14①y至元2000 Peipei Shang School of Mathematical Sciences shang@tongji.edu.Ordinary Differential Equations
Remark A nth-order ODE (An ODE of order n) can always be written in the general form F(x, y, y 0 ,··· , y (n) ) = 0 (2) here F is a given function of n+2 variables ; y is the unknown function; x is the independent variable. (2) is called implicit equation. Peipei Shang School of Mathematical Sciences shang@tongji.edu.cn Ordinary Differential Equations
Definition (Linear and Nonlinear Ordinary Differential Equa- tions） The general nth-order equation F(x,,)=0is linear if the function F is a first-degree polynomial in y,y,) 4口10y至，1元，3000 Peipei Shang School of Mathematical Sciences shang tongji.edu.Ordinary Differential Equations
Definition (Linear and Nonlinear Ordinary Differential Equations) The general nth-order equation F(x, y, y 0 ,··· , y (n) ) = 0 is linear if the function F is a first-degree polynomial in y, y 0 ,··· , y (n) . The general nth-order linear ODE is of the form a0(x)y (n) +a1(x)y (n−1) +···+an−1(x)y 0 +an(x)y = f(x), where a0,a1,··· ,an and f are given functions with a0 6≡ 0. An equation that is not linear is said to be a nonlinear equation. Peipei Shang School of Mathematical Sciences shang@tongji.edu.cn Ordinary Differential Equations