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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 Guessing. When P ( t ) is a constant matrix, the form of the particular solution, ~v ( t ), may be like that of the nonhomogeneous term, ~g ( t ). Subtleties not seen before with the method of undetermined coefficients can arise when applying this method to systems. Example. Find a particular solution of the nonhomogeneous linear system ~x = A~x + ~g ( t ) where A = 1 1 4 1 and ~g ( t ) = 2 e t- e t = 2- 1 e t . The initial guess of the form of a particular solution for this system is ~v ( t ) = ~e t , the substitution of which in the system gives ~e t = A~e t + 2- 1 e t A~- ~ =- 2- 1 ( A- I ) ~ =- 2 1 0 1 4 0 1 2 =- 2 1 ~ = 1 / 4- 2 . A particular solution is therefore v ( t ) = (1 / 4)- 2 e t = (1 / 4) e t...
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