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# Note for MATH 3321 at UH

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Date Created: 02/06/15
62 Systems of Linear Differential Equations Introduction Up to this point the entries in a vector or matrix have been real numbers In this section and in the following sections we will be dealing with vectors and matrices whose entries are functions A vector whose components are functions is called a vectorvalued function or vector functionl Similarly a matrix whose entries are functions is called a matrix function The operations of vector and matrix addition multiplication by a number and matrix multipli cation for vector and matrix functions are exactly as de ned in Chapter 5 so there is nothing new in terms of arithmetic However there are operations on functions other than arithmetic operations elgl limits differentiation and integration that we have to de ne for vector and matrix functions These operation from calculus are de ned in a natural way Let Vt f1 t f2 t l l l fn be a vector function whose components are de ned on an interval It Limit Let c E It If lim fit 1 exists for i 12 l l in then 170 mm 7 grim mm mom a1 a2 an Limits of vector functions are calculated componentwiser Derivative lf f1 f2 Hi fn are differentiable on I then V is differentiable on I and V UN f2 t7 film That is V is the vector function whose components are the derivatives of the compo nents of V Integral Since differentiation of vector functions is done componentwise integration must also be componentwiser That is vtdtltf1tdtf2tdt fntdtgtl Limits differentiation and integration of matrix functions are done in exactly the same way 7 componentwise Systems of Linear differential Equations Consider the thirdorder linear differential equation y WM WW WM 7 f0 where p q 7 f are continuous functions on some interval It Solving the equation for y we get u 7Tty 7 409 7 WM W 237 Introduce new dependent variables 11 12 13 as follows 11 y 12 1 1 y rs 1 2 y Then 9 13 Tt11 4t12 PWIS f and the thirdorder equation can be written equivalently as the system of three rstorder equations I1 I2 12 I3 13 7Ttzl 7 qtzg 7 ptzg Note This system is just a very special case of the general system of three7 rstorder differential equations 11 a11tzl a12tzg a13tzgt b1t 12 a21tzl a22tzg a23tzgt 122t 13 a31t11 a32t12 a33tIst 113t Example 1 a Consider the thirdorder nonhomgeneous equation y 7 y 7 8y 12y 26 Solving the equation for y we have ym 712y 8y y 2amp3 Let 11 y 1 1 12 y 7 1 2 zg y Then m y 1g 71211812132 t and the equation converts to the equivalent system 11 12 2 13 13 71211812zg26t b Consider the secondorder homogeneous equation t2yH 7 ty 7 3y 0 Solving this equation for y we get 3 l H y gyn To convert this equation to an equivalent system7 we let 11 y 11 12 y Then we have 1 I2 3 z 711 712 2 t2 t 238 Which is just a special case of the general system of two rstorder differential equations 11 a11tzl a12tzg b1t 12 a21tzl a22tzg 122 t General Theory Let a11t7 a12t7 m a1nt7 a21t7 m annt7 121t7 122t7 m bnt be continuous functions on some interval If The system of n rstorder differential equations 11 a110311 a12tI2 a1nt1nt 5175 12 a210311 a22tI2 a2nt1nt 5275 S N mm m4mm m is called a rstorder linear di ereritial systeml The system S is homogeneous if b1t E b2t E E bnt E 0 on If S is nonhomogeneous if the functions bit are not all identically zero on I that is7 if there is at least one point a E I and at least one function bit such that 12 a f 0 Let At be the n gtltn matrix a11t a12t a1nt Ag a21t a22t a2nt an1t an t am t and let x and b be the vectors 11 b1 12 122 x 7 b M M Then S can be Written in the vectormatrix form A xh S The matrix At is called the matrix of coe cients or the coe cient matrix Example 2 The vectormatrix form of the system in Example 1 a is a nonhomogeneous systeml The vectormatrix form of the system in Example 1b is 7 0 1 11 0 7 0 1 11 X7 321 2 0 321 2 a homogeneous systeml A solution of the linear differential system S is a differentiable vector function 11 12 Ina that satis es S on the interval 11 3 Example 3 Verify that V lt 372 gt is a solution of the homogeneous system 7 0 1 11 X7 32 1 12 of Example 1 SOLUTION VL 3 7 32 l 0 1 3 7 32 7 6 7 321 32 7 32 7 6 V is a solution 62 1 e 2 Example 4 Verify that V 262 e is a solution of the nonhomogeneous system 462 l e 2 0 1 0 0 X 0 0 1 X 0 712 8 1 26 of Example 1 a 240 SOL UTION rt NlD NlD NlH m m m rt rt Nlb Nlb NlH m m m rt rt rt m 0 ct H s rt OO O O to m 0 ct mmm rt rt to m rt g m 0 ct Nlb Nlb NlH rt O mmm s O NlD NlD NlH 4 2t 26 H0 00 OH o leNlHNlH r l mama M mm mm e 00 H moo OOOH HHO 00 m m a l a N m a ct rt mmm H g m 0 ct Nlb Nlb NlH e V is a solution THEOREM 1 Existence and Uniqueness Theorem Let a be any point on the interval 1 and let a1 12 H i an be any n real numbers Then the initialvalue problem a1 a2 x Atxbt xa an has a unique solution Exercises 62 Convert the differential equation into a system of rstorder equations 1 y 7 tyl3y sin 2t 2 yHy2e 2ti 3 y7yyet 4 my 6y ky cos At m7 5 k A are constants ln Exercises 5 8 a matrix A and a vector b are given Write the system of equations corresponding to X AtX b 241 Alttgt7lt gt7blttgt7ltgjjgt gtblt ltt gt 2 3 71 e At 2 0 1 bt7 2w 1 2 3 0 e2 t2 37 t71 0 At 72 t72 t bt 0 2t 3 t 1 101 111 121 13 1 g 15 11 7211 12 sin t 12 1173127200st Ill girl 7 62t12 12 e tzl 7 36 2t 11 211123133 12 1173127200st 13 2117I24zgt 2 11 tzlz27tzs3 12 7362 213 7 2672i 13 211 7212 413 1 Verify that u gt is a solution of the system in Example 1 gt t 7 te2t 2t 2t 2t 462 4te2t the system in Example 1 a rt 1 2 3t rt 1 Verify that u is a solution of the system in Example 1 e Se Nlb Nlb NlH m m m rt a 1 Verify that W is a solution of the homogeneous system associated With 242 isint icost72sint x i 2 1 x 0 7 73 2 2sint 16 Verify that V lt gt is a solution of the system 726 2 17 Verify that V 0 is a solution of the system 3672 1 73 2 x 0 71 0 x 0 71 72 243

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