ࡱ> /1.pM Nbjbj== (JWW3l4444444H 8 l$<H#Dl"TVVVS<<!#$$ 'E#4E#n44Z#nnn44TnTnn44` %rH8 \<p#0#'p'nHH44443.7 Nonhomogeneous Systems of Differential Equations The general systems of differential equations with constant coefficients can be written as Pi1(D) x1 + Pi2(D) x2 + . . . + Pin(D) xn = fi(t), 1 ( i ( n (3.7-1) The above system can be written in the compact matrix form P(D)x = f with the following definitions of matrix operator P(D), vector x, and vector f P(D) =  EMBED Equation.3 , x =  EMBED Equation.3 , and f =  EMBED Equation.3  The following is an example of a general system of differential equations with constant coefficients.  EMBED Equation.3  + 2x +  EMBED Equation.3  + 6y = 2et 2 EMBED Equation.3  + 3x + 3 EMBED Equation.3  + 8y = (1 The system can be rearranged to  EMBED Equation.3 x +  EMBED Equation.3 y = 2et  EMBED Equation.3 x +  EMBED Equation.3 y = (1 In terms of the differential operator (D + 2)x + (D + 6)y = 2et ( P11(D)x + P12(D)y = 2et (2D + 3)x + (3D + 8)y = (1 ( P21(D)x + P22(D)y= (1 Consider the system (3.7-1) in matrix form P(D)x = f(t) (3.7-1) The associated homogeneous system of equations is P(D)x = 0 (3.7-2) The complete solution of (3.7-2) is called a complementary function of (3.7-1) and any particular solution of (3.7-1) is called a particular integral of that system. To find a complementary function, we assume solutions of the homogeneous system exist in the form x = ke(t Since Dr(e(t) = (re(t, we obtain P(()e(tk = 0 or P(()k = 0 (3.7-3) Eq. (3.7-3) will have a nontrivial solution if |P(()| = 0 (3.7-4) This equation is called the characteristic equation of both the algebraic system (3.7-4) and the original differential system (3.7-2). For each root (i of the characteristic equation (3.7-4) there will be a solution vector ki of (3.7-3) determined to within an arbitrary scalar vector ci. For N distinct real roots (1, (2, . . ., (N. x = c1k1 EMBED Equation.3  + c2k2 EMBED Equation.3  + . . . + cNkN EMBED Equation.3  Example 3.7-1. Find a complete solution of the following first order system. (D + 1)x + (D + 2)y + (D + 3)z = (e(t (D + 2)x + (D + 3)y + (2D + 3)z = e(t (3.7-5) (4D + 6)x + (5D + 4)y + (20D ( 12)z = 7e(t Solution The solution  EMBED Equation.3 =  EMBED Equation.3 e(t of the homogeneous system P(D)x = 0 exists if and only if (( + 1)a + (( + 2)b + (( + 3)c = 0 (( + 2)a + (( + 3)b + (2( + 3)c = 0 (4( + 6)a + (5( + 4)b + (20( ( 12)c = 0 The characteristic equation of this system is  EMBED Equation.3  = (( ( 1) (( ( 2) (( ( 3) = 0 For ( = 1 2a + 3b + 4c = 0 3a + 4b + 5c = 0 (  EMBED Equation.3  = d1 EMBED Equation.3  = c1 EMBED Equation.3  10a + 9b + 8c = 0 For ( = 2 3a + 4b + 5c = 0 4a + 5b + 7c = 0 (  EMBED Equation.3  = d2 EMBED Equation.3  = c2 EMBED Equation.3  14a + 14b + 28c = 0 For ( = 3 4a + 5b + 6c = 0 5a + 6b + 9c = 0 (  EMBED Equation.3  = d3 EMBED Equation.3  = c3 EMBED Equation.3  18a + 19b + 48c = 0 A complementary solution of (3.7-5) is therefore  EMBED Equation.3 = c1 EMBED Equation.3 et + c2 EMBED Equation.3 e2t + c3 EMBED Equation.3 e3t We now need to find a particular integral of the differential system (3.7-5). We choose a trial solution v =  EMBED Equation.3 e-t Substituting v into (3.7-5), collecting terms, and canceling a factor of e-t 4a + 5b + 6c = 0 5a + 6b + 9c = 0 (  EMBED Equation.3  =  EMBED Equation.3  5a + 6b + 9c = 0 Hence a particular integral of (3.7-5) is  EMBED Equation.3 =  EMBED Equation.3 e-t A complete solution of the original system is x = c1 EMBED Equation.3 et + c2 EMBED Equation.3 e2t + c3 EMBED Equation.3 e3t +  EMBED Equation.3 e-t The constants c1, c2, and c3 can be determined from the initial conditions x(0) = 1, y(0) = 1, and z(0) = 1.  EMBED Equation.3  EMBED Equation.3  =  EMBED Equation.3 (  EMBED Equation.3  =  EMBED Equation.3  The solution can be verified with MATLAB. >>[x1,x2,x3]=dsolve('Dx+x+Dy+2*y+Dz+3*z=-exp(-t),Dx+2*x+Dy+3*y+2*Dz+3*z=exp(-t), 4*Dx+6*x+5*Dy+4*y+20*Dz-12*z=7*exp(-t)','x(0)=1,y(0)=1,z(0)=1') x1 = 11/2*exp(t)+24*exp(2*t)-63/2*exp(3*t)+3*exp(-t) x2 = -8*exp(2*t)-11*exp(t)+21*exp(3*t)-exp(-t) x3 = -8*exp(2*t)+11/2*exp(t)+7/2*exp(3*t) PAGE  PAGE 83 5!"UVWXbcpquvwxy}~ jiEHUj0A UV jEHUjvRA CJOJQJUV jU j 6H*]H*6]5\M56stu:;#$_` $ h8a$ $ h8 a$3M=>QRSTX\]pqrswx}~Μun jEHUj1xRA CJOJQJUV j- jEHUjwRA CJOJQJUV] j- EHUjwRA CJOJQJUVH*6] jE EHUjwRA CJOJQJUV6 j]EHUjwRA CJOJQJUV jU jEHUjvRA CJOJQJUV+!"#$&':;<=ABUVWXY\]H* j j- jEHUjxRA CJOJQJUV jEHUjxRA CJOJQJUV5\H* jEEHUjYxRA CJOJQJUV6] jUB!"9:lm   ( $ h8a$ $ h8 a$$%&'(),-./opqrstwx      F G H N O d e f h i  jlH* jlH*5\ j-H*6] jZ( ) w x / 0 Y Z $ !8P @a$$ h8p@ a$$ h8p@ a$$ !P @a$$ h8&d P a$ $ h8 a$ $ h8a$i j s t u z ~  3 4 9 ; > ? 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