ChewMA1506 (2015-16SEM2) Ch7 - MA1506 Mathematics II...

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MA1506 Mathematics II Chapter 7 Systems of First Order ODEs 1 Chew T S MA1506-15 Chapter 7
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7.1 Solving Linear System of ODEs a,b,c,d constants How to solve i.e, We shall look at an old problem , which is related to our new problem, to get ideas to solve our new problem 2 Chew T S MA1506-15 Chapter 7 dx ax by dt dy cx dy dt x a b x d y c d y dt      
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An old problem Consider Zero function is a solution However we are interested in nonzero solutions We know that is the general solution , where is any constant If we want nonzero solutions, then we assume that is nonzero constant 7.1 Solving System of ODEs 3 Chew T S MA1506-15 Chapter 7 ( ) 2 ( ) dx t x t dt ( ) 0 x t 2 0 ( ) t x t x e 0 x 0 x
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So solutions of the old problem are of the form From this old problem, we may guess that solutions of are of the form which can be written as Solutions of systems of ODE 7.1 Solving System of ODEs 4 Chew T S MA1506-15 Chapter 7 0 ( ) t x t x e 0 ( ) t x t x e 0 ( ) t y t y e 0 0 ( ) ( ) t x x t e y y t x a b x d y c d y dt      
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Now we shall find How to find? First substitute into the given ODE, Note that 7.1 Solving System of ODEs 5 Chew T S MA1506-15 Chapter 7 0 0 , , x y 0 0 ( ) ( ) t x x t e y y t 0 0 ( ) ( ) t x x t d e y y t dt x a b x d y c d y dt      
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where Let We get number 2x2 matrix We can not apply cancellation rule to the above equality 7.1 Solving System of ODEs 6 Chew T S MA1506-15 Chapter 7 0 0 0 0 t t x x e B e y y a b B c d 0 0 0 x u y 0 0 u Bu
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Rewrite as where 7.1 Solving System of ODEs 7 Chew T S MA1506-15 Chapter 7 Hence we have We want nonzero solns of the given ODE, so we want to be nonzero vector from Chapter 5, Section 5.5. Therefore 0 0 u Bu 2 0 0 I u Bu 2 1 0 0 1 I 2 0 ( ) 0 B I u 0 u 2 det( ) 0 B I
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Before we look at We shall look at the following equality carefully This equality can be written as eigenvector eigenvalue So we are looking for eigenvalues and eigenvectors of B 7.1 Solving System of ODEs 8 Chew T S MA1506-15 Chapter 7 2 det( ) 0 B I 2 0 ( ) 0 B I u 0 0 Bu u
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Hence Thus Recall Eigenvalue 7.1 Solving System of ODEs 9 Chew T S MA1506-15 Chapter 7 2 det( ) det a b B I c d 0 ( )( ) 0 a d bc 2 1 ( ) 4( ) 2 a d a d ad bc 2 1 ( ) 4(det ) 2 Tr B Tr B B a b B c d 2 ( ) ( ) 0 a d ad bc
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There are three cases Case 1 : Two distinct real roots (eigenvalues) i.e., Suppose two distinct real eigenvalues are
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