Page 322 - Advanced engineering mathematics

302 CHAPTER 10 Systems of Linear Differential Equations

THEOREM 10.4

Let be a fundamental matrix for the homogeneous system X = AX.Let p be any par-

ticular solution of the nonhomogeneous system X = AX + G. Then every solution of the

nonhomogeneous system has the form
X = C + p .
For this reason we call
C + p ,

in which C is an n × 1 matrix of arbitrary constants, the general solution of X = AX + G.
We now know what to look for in solving homogeneous and nonhomogeneous n × n linear
systems.

For the homogeneous system X = AX, form a fundamental matrix whose columns are

n linearly independent solutions. The general solution is X = C.

For the nonhomogeneous system X = AX + G, first find the general solution C of
the associated homogeneous system X = AX. Then find any particular solution p of the

nonhomogeneous system. The general solution of X = AX + G is X = C + p .
In the next section, we will begin to carry out this strategy for the case that the coefficient
matrix A is constant.

SECTION 10.1 PROBLEMS

In each of Problems 1 through 5, (a) verify that the given 3. x = 3x 1 + 8x 2 , x = x 1 − x 2 ,
1 √ 2 √
functions satisfy the system, (b) write the system in matrix x 1 (t)= 4c 1 e (1+2 3)t + 4c 2 e (1−2 3t) ,
√
√
form X = AX for an appropriate A, (c) write n linearly √ (1+ 3)t √ (1−2 3)6t

x 2 (t)= (−1 + 3)c 1 e + (−1 − 3)c 2 e ,
independent n × 1 matrix solutions 1 ,··· , n , for appro-
x 1 (0)= 2, x 2 (0) = 2
priate n, (d) use the determinant test of Theorem 10.2(2)
to verify that these solutions are linearly independent, (e) 4. x = x 1 − x 2 , x = 4x 1 + 2x 2 ,

2
1
form a fundamental matrix for the system, and (f) use the √ √
x 1 (t)= 2e 3t/2 c 1 cos( 15t/2) + c 2 sin( 15t/2) ,
fundamental matrix to solve the initial value problem.
√ √ √
x 2 (t)= c 1 e 3t/2 −cos( 15t/2) + 15sin( 15t/2)

1. x = 5x 1 + 3x 2 , x = x 1 + 3x 2 , √ √ √

1
2
6t
6t
2t
2t
x 1 (t)= c 1 e + 3c 2 e , x 2 (t) = c 1 e + c 2 e , −c 2 e 3t/2 sin( 15t/2) + 15cos( 15t/2) ,
x 1 (0)= 0, x 2 (0) = 4 x 1 (0)=−2, x 2 (0) = 7
2. x = 2x 1 + x 2 , x =−3x 1 + 6x 2 , 5. x = 5x 1 − 4x 2 + 4x 3 , x = 12x 1 − 11x 2 + 12x 3 ,

1
1
2
2
4t
4t
x 1 (t)= c 1 e cos(t) + c 2 e sin(t) x (t)= 4x 1 − 4x 2 + 5x 3

3
4t t −3t 2t −3t
x 2 (t)= 2c 1 e [cos(t) − sin(t)] x 1 (t)=−c 1 e + c 3 e , x 2 (t) = c 2 e + c 3 e ,
t
4t
+2c 2 e [cos(t) + sin(t)], x 3 (t)= (c 3 − c 1 )e + c 3 e −3t ,
x 1 (0)=−2, x 2 (0) = 1 x 1 (0)= 1, x 2 (0) =−3, x 3 (0) = 5

10.2 Solution of X = AX for Constant A
Now we know what to look for to solve a linear system. We must find n linearly independent
solutions.
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
October 14, 2010 20:32 THM/NEIL Page-302 27410_10_ch10_p295-342