lecture2 - Dierential Equations Linear Algebra Lecture 2...

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Differential Equations & Linear Algebra Lecture 2 Jianli XIE Email: [email protected] Department of Mathematics, SJTU Fall 2010
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Contents Linear equations: Integrating factors
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Contents Linear equations: Integrating factors Separable equations
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Contents Linear equations: Integrating factors Separable equations Method of variation of parameters
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Contents Linear equations: Integrating factors Separable equations Method of variation of parameters Homogeneous equations
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Contents Linear equations: Integrating factors Separable equations Method of variation of parameters Homogeneous equations Bernoulli equations
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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 ) μ ( s ) g ( s ) ds + c .
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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 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 . Multiplying the differential equation by μ ( t ) , we get e - 2 t y ( t ) - 2 e - 2 t y = 4 e - 2 t - te - 2 t , or ( e - 2 t y ) = 4 e - 2 t - te - 2 t .
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Example Integrating both sides, we have e - 2 t y = - 2 e - 2 t + 1
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