Chapter-6 - CHAPTER 6 Linear Systems of Differential...

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564 CHAPTER 6 Linear Systems of Differential Equations 6.1 Theory of Linear DE Systems ± Breaking Out Systems 1. 11 2 21 2 2 4 x xx x =+ =− 2. 22 1 ′ = ′ = −+ 3. 112 2 43 t x e x =++ ′ =− − 4. 12 23 31 2 3 s i n x xx x t ′ = ′ = ′=− + + + ± Checking It Out 5. 13 ⎡⎤ ′ = ⎢⎥ ⎣⎦ GG Substituting () 4 4 t t e t e = u G and 2 2 t t e t e = v G into the given system easily verifies: 44 41 3 4 tt ee = and 3 2 −− ⎤⎡ = ⎥⎢ ⎦⎣ . The fundamental matrix 42 t = X . The general solution of this 2 2 × system tc c x G .
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SECTION 6.1 Theory of Linear DE Systems 565 6. 41 21 ⎡⎤ ′ = ⎢⎥ ⎣⎦ xx GG By substitution, we verify that () 3 3 t t e t e = u G and 2 2 2 t t e t e = v G satisfy the system. The fundamental matrix 32 2 tt ee t = X . The general solution 12 2 tc c ⎤⎡ =+ ⎥⎢ ⎦⎣ x G . 7. 11 ′ = By substitution, we verify that 2 t t e t e = u G and 3 3 2 t t e t e = v G satisfy the system. The fundamental matrix 3 3 22 t = X . The general solution 3 3 c x G . 8. 01 10 ′ = By substitution, we verify that sin cos t t t = u G and cos sin t t t = v G satisfy the system. The fundamental matrix sin cos cos sin t = X . The general solution sin cos cos sin c x G .
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566 CHAPTER 6 Linear Systems of Differential Equations ± Uniqueness in the Phase Plane 9. The direction field of x y ′ = , yx ′=− is shown. We have drawn three distinct trajectories for the six initial conditions ( ) ( ) ( ) 0, 0 xy = ( ) 1, 0 , ( ) 2, 0 , ( ) 3, 0 , () 0,1 , ( ) 0, 2 , ( ) 0, 3 . Note that although the trajectories may (and do) coincide if one starts at a point lying on another, they never cross each other. However, if we plot coordinate ( ) x xt = or ( ) yy t = for these same six initial conditions we get the six intersecting curves shown in the tx and ty planes. Intersecting solutions ( ) = Intersecting solutions ( ) t =
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SECTION 6.1 Theory of Linear DE Systems 567 ± Verification 10. Substituting t t e e ⎡⎤ = ⎢⎥ ⎣⎦ v G into 11 22 12 21 x x x x ⎤⎡ = ⎥⎢ ⎦⎣ yields tt ee −− = , or 2 2 t t e e −= = , which verifies that v G is the solution. ± Third-Order Verification 11. To verify , , , uvw GGG , you should follow the procedure carried out in Problem 10. To show that the vector functions , , are linearly independent, set 123 1 2 3 2 0 0 0 00 t et e ccc c ec e c t e ⎡⎤ ⎡ ⎤ ⎢⎥ ⎢ ⎥ ++ = + + = ⎢ ⎥ ⎣⎦ ⎣ ⎦ or, in scalar form, 23 3 2 13 0 0 0 t ce cte += = .
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This note was uploaded on 10/26/2009 for the course MA 232 taught by Professor Toland during the Fall '08 term at Clarkson University .

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Chapter-6 - CHAPTER 6 Linear Systems of Differential...

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