Example Next,substitution of(6)in(4)gives x()-G(2cne-Scze-)+c+eze5) that is, 0-9e2+ 3 32e5r (7) Thus(6)and(7)constitute the general solution of the system in(3).The given initial conditions imply that x(0)= 3,1 29+39=2 and that y(0)=c1+c2=-1; 4口10,1至，无3000 Linear of Differential Equations
Example Next, substitution of (6) in (4) gives x(t) = 1 6 (2c1e 2t −5c2e −5t ) + 7 6 (c1e 2t +c2e −5t ); that is, x(t) = 3 2 c1e 2t + 1 3 c2e −5t . (7) Thus (6) and (7) constitute the general solution of the system in (3). The given initial conditions imply that x(0) = 3 2 c1 + 1 3 c2 = 2 and that y(0) = c1 +c2 = −1; Linear Systems of Differential Equations
Example these equations are readily solved for c1=2 and c2=-3.Hence the desired solution is x0=3e2-e-,y0=2e2-3e5. 4口14①y至元2000 Linear of Differential Equations
Example these equations are readily solved for c1 = 2 and c2 = −3. Hence the desired solution is x(t) = 3e 2t −e −5t , y(t) = 2e 2t −3e −5t . Linear Systems of Differential Equations
Definition (First-Order System) A system of differential equations for unknown functions x1,x2,.,xn that is of the form =fi,x1,,n dt dx d 2=fh,2,…n (8) =f,2n d where the functions fi are given,is called a first-order system. 口10yt至1元3000 Linear of Differential Equations
Definition (First-Order System) A system of differential equations for unknown functions x1, x2,··· , xn that is of the form dx1 dt = f1(t, x1, x2,··· , xn), dx2 dt = f2(t, x1, x2,··· , xn), (8) ······ dxn dt = fn(t, x1, x2,··· , xn), where the functions fi are given, is called a first-order system. Linear Systems of Differential Equations
Definition A solution of the system (8)is an ordered set of functions (x1,x2,.)that satisfies the system on some interval. 4口14①yt至2000 Linear of Differential Equations
Definition A solution of the system (8) is an ordered set of functions (x1, x2,··· , xn) that satisfies the system on some interval. Definition A set of auxiliary conditions of the form x1(t0) = k1, x2(t0) = k2,··· , xn(t0) = kn, (9) where the ki are given numbers, is called a set of initial conditions for the system. The system together with the initial conditions is called an initial value problem. Linear Systems of Differential Equations
Definition A solution of the system (8)is an ordered set of functions (x1,x2,..)that satisfies the system on some interval. Definition A set of auxiliary conditions of the form x1(t0)=k1,2(to)=k2,·,xn(t0)=kn, (9) where the ki are given numbers,is called a set of initial con- ditions for the system. 4口14①yt至2000 Linear of Differential Equations
Definition A solution of the system (8) is an ordered set of functions (x1, x2,··· , xn) that satisfies the system on some interval. Definition A set of auxiliary conditions of the form x1(t0) = k1, x2(t0) = k2,··· , xn(t0) = kn, (9) where the ki are given numbers, is called a set of initial conditions for the system. The system together with the initial conditions is called an initial value problem. Linear Systems of Differential Equations