{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

M334Lec35

# M334Lec35 - Math 334 Lecture#35 Â 7.9 Nonhomogeneous Linear...

This preview shows pages 1–2. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Math 334 Lecture #35 Â§ 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 ) = Î¨( t ) ~ c + ~v ( t ) where Î¨( 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 ODEâ€™s, ~ 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....
View Full Document

{[ snackBarMessage ]}

### Page1 / 3

M334Lec35 - Math 334 Lecture#35 Â 7.9 Nonhomogeneous Linear...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online