lecture10 - Differential Equations & Linear...

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Unformatted text preview: Differential Equations & Linear Algebra Lecture 10 Jianli XIE Email: xjl@sjtu.edu.cn Department of Mathematics, SJTU Fall 2010 Contents Homogeneous equations with constant coefficients Case I: real and unequal roots Case II: complex roots Case III: repeated roots Contents Homogeneous equations with constant coefficients Case I: real and unequal roots Case II: complex roots Case III: repeated roots Nonhomogeneous equations: the method of undetermined coefficients Case I: g ( t ) = P n ( t ) Case II: g ( t ) = P n ( t ) e αt Case III: g ( t ) = P n ( t ) e αt sin βt or P n ( t ) e αt cos βt Contents Homogeneous equations with constant coefficients Case I: real and unequal roots Case II: complex roots Case III: repeated roots Nonhomogeneous equations: the method of undetermined coefficients Case I: g ( t ) = P n ( t ) Case II: g ( t ) = P n ( t ) e αt Case III: g ( t ) = P n ( t ) e αt sin βt or P n ( t ) e αt cos βt Method of variation of parameters Contents Homogeneous equations with constant coefficients Case I: real and unequal roots Case II: complex roots Case III: repeated roots Nonhomogeneous equations: the method of undetermined coefficients Case I: g ( t ) = P n ( t ) Case II: g ( t ) = P n ( t ) e αt Case III: g ( t ) = P n ( t ) e αt sin βt or P n ( t ) e αt cos βt Method of variation of parameters Reduction of order Equations with constant coefficients Consider the n th order linear homogeneous differential equation L [ y ] = a y ( n ) + a 1 y ( n- 1) + ··· + a n- 1 y + a n y = 0 , (1) where a , a 1 , ··· , a n are real constants. From our knowledge of second order linear equations with constant coefficients, it is natural to anticipate that y = e rt is a solution of equation (9) for suitable values of r . In fact, L [ e rt ] = e rt ( a r n + ··· + a n- 1 r + a n ) = e rt Z ( r ) , (2) where Z ( r ) = a r n + a 1 r n- 1 + ··· + a n- 1 r + a n . (3) Characteristic equation For those values of r for which Z ( r ) = 0 , it follows that L [ e rt ] = 0 and y = e rt is a solution of equation (9). The polynomial Z ( r ) is called the characteristic polynomial , and the equation Z ( r ) = 0 is the characteristic equation of the differential equation (9). A polynomial of degree n has n zeros, say r 1 , r 2 , ··· , r n , some of which may be equal and some of which may be complex conjugates; hence we can write the characteristic polynomial in the form Z ( r ) = a ( r- r 1 )( r- r 2 ) ··· ( r- r n ) . (4) Real and unequal roots If the roots of the characteristic equation are real and no two are equal, then we have n distinct solutions e r 1 t , e r 2 t , ··· , e r n t of equation (9). If these functions are linearly independent, then the general solution of equation (9) is y = c 1 e r 1 t + c 2 e r 2 t + ··· + c n e r n t . (5) One way to establish the linear independence of e r 1 t , e r 2 t , ··· , e r n t is to evaluate their Wronskian: W = e r 1 t e r 2 t ··· e r n t r 1 e r 1 t r 2 e r 2 t ··· r n...
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lecture10 - Differential Equations & Linear...

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