Example The temperature u=u(x,t)of a long thin uniform rod satisfies ∂u,a2u 01-k 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. 4口14①y至元2000 Ordinary Differential Equations
Example 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. Ordinary Differential Equations
Example Concerning the motion of a particle(with mass m)in terms of the force F acting on it,we denote by x(t)the position of the particle at time t,then the velocity is dx v= dr Its acceleration a(t)is a- From Newton's second law of motion ma(t)=F(t) dx F(t) d2= m 4口10y至，无2000 Ordinary Differential Equations
Example Concerning the motion of a particle (with mass m) in terms of the force F acting on it, we denote by x(t) the position of the particle at time t, then the velocity is v = dx dt Its acceleration a(t) is a = dv dt = d 2 x dt2 From Newton’s second law of motion ma(t) = F(t) d 2 x dt2 = F(t) m . Ordinary Differential Equations
Terminology Definition The order of a differential equation is the highest order deriva- tive that appears in the differential equation. 4日10y至，1元3000 Ordinary Differential Equations
Terminology Definition The order of a differential equation is the highest order derivative that appears in the differential equation. Example (1) y 0 +2xy = 0; (first order) (2) xy 00 +y 0 = x 2 ; (second order) (3) y 000 −x(y 0 ) 3 +y = 0; (third order) Ordinary Differential Equations
Terminology Definition The order of a differential equation is the highest order deriva- tive that appears in the differential equation. Example (1)y+2xy=0: 4日10y4至，1无2000 Ordinary Differential Equations
Terminology Definition The order of a differential equation is the highest order derivative that appears in the differential equation. Example (1) y 0 +2xy = 0; (first order) (2) xy 00 +y 0 = x 2 ; (second order) (3) y 000 −x(y 0 ) 3 +y = 0; (third order) Ordinary Differential Equations
Terminology Definition The order of a differential equation is the highest order deriva- tive that appears in the differential equation. Example (1)y+2xy=0;(first order) 4口10y4至，元2000 Ordinary Differential Equations
Terminology Definition The order of a differential equation is the highest order derivative that appears in the differential equation. Example (1) y 0 +2xy = 0; (first order) (2) xy 00 +y 0 = x 2 ; (second order) (3) y 000 −x(y 0 ) 3 +y = 0; (third order) Ordinary Differential Equations