MATH-M 344 Chapter 7: M344 7.5 Notes (Feb. 1)
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M344 section 7. 5 notes- homogeneous linear systems with constant coefficients. Focus on systems of form =, where is constant matrix (assume all elements real unless stated. If =(cid:883) (vectors have 1 element, matrix is scalar ), system reduces to single first order equation = with solution =(cid:1857) For equations, equilibrium solutions found by solving =(cid:2777) (homogeneous matrix equation) Assume det (cid:882), so =(cid:2777) is the only equilibrium solution (does not change with time) Question: do other solutions approach or diverge from equilibrium solution as (is =(cid:2777) Find the general solution of the system =[(cid:884: notice that is a diagonal matrix (cid:882) (cid:885)][(cid:2869)(cid:2870)] (cid:4666)(cid:2869) ,(cid:2870) (cid:4667)=[(cid:884) (cid:882) (cid:4666)(cid:2869) ,(cid:2870) (cid:4667)=(cid:4666)(cid:884)(cid:2869), (cid:885)(cid:2870)(cid:4667) {(cid:2869) =(cid:884)(cid:2869) (cid:2870) = (cid:885)(cid:2870) each equation involves 1 unknown; can solve individually: (cid:2869) =(cid:884)(cid:2869) (cid:3117) =(cid:884)(cid:2869) (cid:2869)(cid:2870)(cid:3117)(cid:1856)(cid:2869)=(cid:883) (cid:1856) (cid:2869)(cid:2870) (cid:2869)(cid:3117)(cid:1856)(cid:2869)=+(cid:1855)(cid:2869) (cid:2869)(cid:2870)ln|(cid:2869)|=+(cid:1855)(cid:2869) ln|(cid:2869)|=(cid:884)+(cid:1855)(cid:2869) (cid:2869)=(cid:1855)(cid:2869)(cid:1857)(cid:2870, (cid:2870) = (cid:885)(cid:2870) (cid:3118) = (cid:885)(cid:2870) (cid:2869)(cid:2871)ln|(cid:2870)|=+(cid:1855)(cid:2870) ln|(cid:2870)|= (cid:885)+(cid:1855)(cid:2870) (cid:2870)=(cid:1855)(cid:2870)(cid:1857) (cid:2871: write solution in vector form: (cid:4666)(cid:4667)=(cid:4666)(cid:1855)(cid:2869)(cid:1857)(cid:2870),(cid:1855)(cid:2870)(cid:1857) (cid:2871)(cid:4667)
