Definition (Linear and Nonlinear Ordinary Differential Equa- tions The general nth-order equation F(x,y,y,()=0 is linear if the function F is a first-degree polynomial in y,,) The general nth-order linear ODE is of the form ao(x)y(m)+ai(x)y(-1)+...+an-1(x)y/+an(x)y=f(x), where ao,a1,...,an and f are given functions with ao0.An equation that is not linear is said to be a nonlinear equation. 4口14①y4至2000 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
Example (1)y+2y=0: 4日10y4至，1无2000 Peipei Shang School of Mathematical Sciences shang@tongji.edu.Ordinary Differential Equations
Example (1) y 0 +2xy = 0; (linear) (2) xy 00 +y 0 = x 2 ; ( linear) (3) y 000 −x(y 0 ) 3 +y = 0; (nonlinear) (4) d 2 y dt 2 +y = cos 2t. (linear) Peipei Shang School of Mathematical Sciences shang@tongji.edu.cn Ordinary Differential Equations
Example (1)y+2xy=0;(linear) 4日10y4至，1无2000 Peipei Shang School of Mathematical Sciences shang@tongji.edu.Ordinary Differential Equations
Example (1) y 0 +2xy = 0; (linear) (2) xy 00 +y 0 = x 2 ; ( linear) (3) y 000 −x(y 0 ) 3 +y = 0; (nonlinear) (4) d 2 y dt 2 +y = cos 2t. (linear) Peipei Shang School of Mathematical Sciences shang@tongji.edu.cn Ordinary Differential Equations
Example (1)y+2xy=0;(linear) (2)"+y=x2; 4日10y4至，1无2000 Peipei Shang School of Mathematical Sciences shang@tongji.edu.Ordinary Differential Equations
Example (1) y 0 +2xy = 0; (linear) (2) xy00 +y 0 = x 2 ; ( linear) (3) y 000 −x(y 0 ) 3 +y = 0; (nonlinear) (4) d 2 y dt 2 +y = cos 2t. (linear) Peipei Shang School of Mathematical Sciences shang@tongji.edu.cn Ordinary Differential Equations
Example (1)y+2xy=0;(linear) (2)x"+y=x2;（linear) 4日10y4至，1无2000 Peipei Shang School of Mathematical Sciences shang@tongji.edu.Ordinary Differential Equations
Example (1) y 0 +2xy = 0; (linear) (2) xy00 +y 0 = x 2 ; ( linear) (3) y 000 −x(y 0 ) 3 +y = 0; (nonlinear) (4) d 2 y dt 2 +y = cos 2t. (linear) Peipei Shang School of Mathematical Sciences shang@tongji.edu.cn Ordinary Differential Equations