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# Class Note for MATH 3321 at UH

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Date Created: 02/06/15
THEOREM 4 Let V1 V2 Vn be n ncomponent vector functions de ned on an interval I If the vectors are linearly dependent then U11 U12 39 39 39 Uin U21 U22 39 39 39 U27 E 0 on I U711 U712 39 39 39 39Unn The determinant in Theorem 4 is called the Wronskz39zm of the vector functions V1 V2 Vn Example 1 The vector functions t3 til ult3t2gt and Vlt7t72gt are solutions of the homogeneous system in Example 3 Section 62 Their Wronskian is 3 r1 W 747 I 32 4 2 The vector functions 62 673 mm V1 2e2t V2 73E V3 62 2te2 4 2t 96 3t 462t 4t 2t are solutions of the homogeneous system 0 l 0 X 0 0 l X 712 8 1 Their Wronskian is 62 673 mm 2e2t 7367 2t 2te2t 25Et 462 96 3t 4 2t 4t 2t THEOREM 5 Let V1 V2 Vn be n solutions of Exactly one of the following holds 1 WV1 V2 Vn t E 0 on I and the solutions are linearly dependent 2 WV1 V2 Vn t f 0 for all t E I and the solutions are linearly independent It is easy to construct sets of n linearly independent solutions of Simply pick any point a E I and any nonsingular n X n matrix A Let 11 be the rst column of A 12 the second column of A and so on Then let V1 be the solution of such that V1a 11 let V2 be the solution of such that V2a a2 and let Vn be the solution of such that Vn an The existence and uniqueness theorem guarantees the existence of these solutions Now WV1V2 Vna detA y 0 245 Therefore Wt 0 for all t E I and the solutions are linearly independent A particularly nice set of n linearly independent solutions is obtained by choosing A In the identity matrix THEOREM 6 Let V1 V2 Hi Vn be it linearly independent solutions of Let u be any solution of Then there exists a unique set of constants Cl 52 i i i on such that u 61V1CQV2 cnvni That is every solution of can be Written as a unique linear combination of V1 V2 H i Vni DEFINITION 2 A set V1 V2 H i Vn of n linearly independent solutions of is called a fundamental set of solutions A fundamental set of solutions is also called a solution basis for If V1 V2 H i Vn is a fundamental set of solutions of H then the n X n matrix U119 U129 390an VO v21t v22 v2nt Un t 1M2 t 39 Unn the vectors V1 V2i i i Vn are the columns of V is called a fundamental matrix for DEFINITION 3 Let V1 V2 H i Vn be a fundamental set of solutions of Then x clvl C2V2 cnvn Where Cl 52 i i i on are arbitrary constants is the general solution of t3 til ult3t2gt and Vlt7t72gt are a fundamental set of solutions of 7 0 1 11 x 7 3t2 1 12 wt lt 3 ii is a fundamental matrix for the systemi The vector function Example 2 The vectors The matrix x clu 62V is the general solution of the systemi Exercises 63 Determine Whether or not the vector functions are linearly dependenti 2t71 7t1 Lu 7 V it 2t 246 10 11 lt cost gt lt sint gt u 7 V sint cost 7 t2 72t4t2 u 7 V 775 2t tet et u 7 V t 1 277 t 27 u t 7 V 71 7 W t7 2 72 2 2 cost cost 0 u sin 7 7 V 0 7 W cost 0 sint sint 6t 76t 0 u fat 7 V 26 7 W 6t 6t 76t 0 2775 tl t u 7 V 7 W t 72 t2 et 0 0 u 7 V 7 W 0 0 et cos t7r4 cost sin t 0 0 0 u 7 7 V 7 7 W 7 0 0 0 0 6t 6t eZt d u 67 an 7lt a Show that u7 V are a fundamental set of solutions of the systemi b Let V be the corresponding fundamental matrixi Show that V AV C Give the general solution of the systemi d Find the solution of the system that satis es x0 lt 3 247 12 Let V be the matrix function V6 Cos 2t s1n 2t s1n 2t 7 cos 2t a Verify that V is a fundamental matrix for the system 7072 2 0 b Find the solution of the system that satis es x0 lt 3 13 Let V be the matrix function 0 476quot 6quot Vt 1 equot 0 1 0 0 a Verify that V is a fundamental matrix for the system 71 4 74 x 0 71 1 x 0 0 0 0 b Find the solution of the system that satis es x0 1 2 248 63 Homogeneous Systems In this section we give the basic theory for linear homogeneous systems This theory is simply a repetition results given in Sections 32 and 617 phrased this time in terms of the system 11 a110311 M20312 a1nt1n 75 12 a210311 M20312 a2nt1n 75 W 1 an1t11 an2t12 amtznt or x Atx 0 0 Note rst that the zero vector zt E 0 is a solution of As before7 this solution 0 is called the trivial solution Of course7 we are interested in nding nontrivial solutions THEOREM 1 If V is a solution of and a is any real number7 then u av is also a solution of any constant multiple of a solution of is a solution of THEOREM 2 If V1 and V2 are solutions of H7 then u V1 V2 is also a solutions of the sum of any two solutions of is a solution of These two theorems can be combined and extended to THEOREM 3 If V17 V27 vk are solutions of H7 and if Cl 52 H ck are real numbers7 then CiVi C2V2 Cka is a solution of any linear combination of solutions of is also a solution of DEFINITION 1 Let v11t v12t vlk t v21t v22t ng t V1 7 V2 7 i i i 7 Vic Un1t on t 11 t be ncomponent vector functions de ned on some interval 1 The vectors are linearly dependent on I if there exist It real numbers Cl 52 ck not all zero7 such that clvlt Cnglttgt ckvkt E 0 on 1 Otherwise the vectors are linearly independent on I 244

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