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

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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 ) = Ψ( t ) c + v ( t ) where Ψ( 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 = Ax + 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 ) = ηe t , the substitution of which in the system gives ηe t = Aηe t + 2 - 1 e t - η = - 2 - 1 ( A - I ) η = - 2 1 0 1 4 0 η 1 η 2 = - 2 1 η = 1 / 4 - 2 .
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A particular solution is therefore v ( t ) = (1 / 4) - 2 e t = (1 / 4) e t - 2 e t .
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