# Lecture 18 - Nonhomogeneous Linear Systems Undetermined...

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Nonhomogeneous Linear Systems Undetermined Coefficients

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Homogeneous Linear Systems y 1 y 2 = a 22 a 12 a 11 a 21 y 0 1 y 0 2 Find Eigenvalues and Eigenvectors λ 1 v 1 and and v 2 λ 2 If Eigenvalues are distinct (real or complex) y 1 y 2 = C 1 C 2 + e λ 1 t v 1 e λ 2 t v 2 If Eigenvalue is repeated = + e λ 1 t v 1 v 3 e λ 1 t t e λ 1 t v 1 C 1 + C 2 y 1 y 2 C 2
Non-Homogeneous Linear Systems y 1 y 2 = a 22 a 12 a 11 a 21 y 0 1 y 0 2 g 1 ( t ) g 2 ( t ) + Remember, general solution takes the form y 1 y 2 = General Solution To Homogeneous Equation + Specific Solution to Nonhomogenous Equation Know how to find this

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Non-Homogeneous Linear Systems y 1 y 2 = a 22 a 12 a 11 a 21 y 0 1 y 0 2 g 1 ( t ) g 2 ( t ) + Remember, general solution takes the form y 1 y 2 = General Solution To Homogeneous Equation + Specific Solution to Nonhomogenous Equation Today we will learn how to find this
Non-Homogeneous Linear Systems y 1 y 2 = a 22 a 12 a 11 a 21 y 0 1 y 0 2 g 1 ( t ) g 2 ( t ) + Several Different Methods We will focus on Undetermined Coefficients

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Method of Undetermined Coefficients y 1 y 2 = y 0 1 y 0 2 + 1 0 1 2 - 1 - 3 = y 1 + y 2 e t e t
Method of Undetermined Coefficients y 1 y 2 = y 0 1 y 0 2 + 1 0 1 2 - 1 - 3 Rewrite Homogeneous Term as sum of Constant Vectors times Functions e t e t

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