The Method of Elimination Example Find the particular solution of the system x=4x-3y,y'=6r-7y (3) that satisfies the initial conditions x(0)=2,y(0)=-1 Solution:If we solve the second equation in(3)for x,we get (4) so that (5) 4口0y4主，元，3000 Linear of Differential Equations
The Method of Elimination Example Find the particular solution of the system x 0 = 4x−3y, y 0 = 6x−7y (3) that satisfies the initial conditions x(0) = 2, y(0) = −1. Solution: If we solve the second equation in (3) for x, we get x = 1 6 y 0 + 7 6 y, (4) so that x 0 = 1 6 y 00 + 7 6 y 0 . (5) Linear Systems of Differential Equations
Example We then substitute these expressions for x andx in the first equation of the system in(3);this yields 6+=4(信+)- which we simplify to y"+3y-10y=0. 0a0 Linear of Differential Equations
Example We then substitute these expressions for x and x 0 in the first equation of the system in (3); this yields 1 6 y 00 + 7 6 y 0 = 4 1 6 y 0 + 7 6 y −3y, which we simplify to y 00 +3y 0 −10y = 0. This second-order linear equation has characteristic equation r 2 +3r −10 = (r −2)(r −5) = 0, so its general solution is y(t) = c1e 2t +c2e −5t . (6) Linear Systems of Differential Equations
Example We then substitute these expressions for x andx in the first equation of the system in(3);this yields +名=4（信日+）- which we simplify to y"+3y-10y=0. This second-order linear equation has characteristic equation 2+3r-10=(r-2)(r-5)=0, so its general solution is y(t)=cle2+c2e-51. (6) Linear of Differential Equations
Example We then substitute these expressions for x and x 0 in the first equation of the system in (3); this yields 1 6 y 00 + 7 6 y 0 = 4 1 6 y 0 + 7 6 y −3y, which we simplify to y 00 +3y 0 −10y = 0. This second-order linear equation has characteristic equation r 2 +3r −10 = (r −2)(r −5) = 0, so its general solution is y(t) = c1e 2t +c2e −5t . (6) Linear Systems of Differential Equations
Example Next,substitution of(6)in(4)gives )G(2ce"-Scxe-5)+(cne+ce) that is, 9e2+ 392e (7) 4口14y至，2000 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 Next,substitution of(6)in(4)gives x()-G(2cne-5cze-5)+(ce+cxe-5) that is, 0=e产+ 3cze-5 (7) Thus(6)and(7)constitute the general solution of the system in(3). 4口14y至，2000 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