lecture2 - Differential Equations& Linear Algebra...

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Unformatted text preview: Differential Equations & Linear Algebra Lecture 2 Jianli XIE Email: [email protected] Department of Mathematics, SJTU Fall 2010 Contents Linear equations: Integrating factors Contents Linear equations: Integrating factors Separable equations Contents Linear equations: Integrating factors Separable equations Method of variation of parameters Contents Linear equations: Integrating factors Separable equations Method of variation of parameters Homogeneous equations Contents Linear equations: Integrating factors Separable equations Method of variation of parameters Homogeneous equations Bernoulli equations Integrating factors Consider the first order linear equation dy dt + p ( t ) y = g ( t ) . (1) Multiplying a factor μ ( t ) , we obtain μ ( t ) y + μ ( t ) p ( t ) y = μ ( t ) g ( t ) . (2) If left-hand side is the derivative of μ ( t ) y ( t ) , that is, μ ( t ) y ( t ) = μ ( t ) p ( t ) y ( t ) (3) then equation (2) can be solved as y ( t ) = 1 μ ( t ) Z μ ( s ) g ( s ) ds + c . Integrating factors The function μ ( t ) satisfying (3) is called an integrating factor . We obtain the simplest possible function μ ( t ) by solving the equation (3): μ ( t ) = e R p ( t ) dt . (4) Example Solve the differential equation dy dt- 2 y = 4- t. Solution. The integrating factor is μ ( t ) = e- 2 t ....
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This note was uploaded on 05/17/2011 for the course ME 255 taught by Professor Jianlixie during the Fall '10 term at Shanghai Jiao Tong University.

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lecture2 - Differential Equations& Linear Algebra...

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