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Unformatted text preview: Math 334 Lecture #35 7.9: Nonhomogeneous Linear Systems Form of General Solution. A general solution of a nonhomogeneous linear system ~x = P ( t ) ~x + ~g ( t ) has the form ~x ( t ) = ( t ) ~ c + ~v ( t ) where ( t ) is a fundamental matrix for the associated homogeneous linear system ~x = P ( t ) ~x , and where ~v ( t ) is a particular solution of the nonhomogeneous linear system. This follows because the difference of any two solutions, ~v ( t ) and ~u ( t ), of the nonohomo- geneous system is a solution of the homogeneous system: ~v ( t )- ~u ( t ) = ~v ( t )- ~u ( t ) = P ( t ) ~v ( t ) + ~g ( t )- P ( t ) ~u ( t )- ~g ( t ) = P ( t ) ~v ( t )- ~u ( t ) . Finding a Particular Solution Through Diagonalization. When P ( t ) is a con- stant matrix A that is diagonalizable, the invertible linear change of variables ~x = T~ y transforms the nonhomogeneous system ~x = A~x + ~g ( t ) into uncoupled first order non- homogeneous linear ODEs, ~ y = T- 1 ~x = T- 1 A~x + ~g ( t ) = T- 1 A~x + T- 1 ~g ( t ) = T- 1 AT~ y + T- 1 ~g ( t ) = D~ y + T- 1 ~g ( t ) , each of which is readily solved; converting the general solution ~ y ( t ) back to the original coordinates ~x ( t ) = T~ y ( t ) gives the fundamental matrix as well as a particular solution....
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This note was uploaded on 03/08/2011 for the course MATH 334 taught by Professor Smith during the Spring '11 term at Vanderbilt.
- Spring '11
- Linear Systems