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lecture19 - Dierential Equations Linear Algebra Lecture 19...

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Differential Equations & Linear Algebra Lecture 19 Jianli XIE Email: [email protected] Department of Mathematics, SJTU Fall 2010
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Contents Systems of first order linear equations General form of systems of first order equations Existence and uniqueness theorems
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Contents Systems of first order linear equations General form of systems of first order equations Existence and uniqueness theorems Matrix functions
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Contents Systems of first order linear equations General form of systems of first order equations Existence and uniqueness theorems Matrix functions Basic theory of systems of first order linear equations
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Introduction There are many physical problems that involve a number of separate elements linked together in some manner. In these cases, the corresponding mathematical problem consists of a system of two or more differential equations, which can always be written as first order equations. To unify the presentation and to solve the systems of differential equations, we will utilize some of the elementary aspects of linear algebra. Systems of ordinary differential equations arise naturally in problems involving several dependent variables each of which is a function of the same single independent variable.
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Notations We will denote the independent variable by t , and will let x 1 , x 2 , x 3 , · · · represent dependent variables that are functions of t . Differentiation with respect to t will be denoted by a prime. Example A predator-prey system. Let x 1 ( t ) and x 2 ( t ) be the numbers of predator and prey, respectively. Without the prey the predators will increase, and without the predator the prey will increase. A simple predator-prey model is x 1 = a 1 x 1 - b 1 x 1 x 2 , x 2 = - a 2 x 2 + b 2 x 1 x 2 . The coefficient a 1 is the birth rate of the predator; a 2 is the death rate of the prey. The x 1 x 2 terms model the interaction of the two populations.
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Higher order equations Another reason why systems of first order equations are important is that equations of higher order can always be transformed into such systems. Example Suppose a spring-mass system is described by the second order differential equation u + 0 . 125 u + u = 0 . Rewrite this equation as a system of first order equations. Solution Let x 1 = u and x 2 = u . Then it follows that x 1 = x 2 and u = x 2 . Then by substituting u, u , u in the differential equation, we obtain x 2 + 0 . 125 x 2 + x 1 = 0 . Thus x 1 and x 2 satisfy the following system of two first order differential equations: x 1 = x 2 , x 2 = - x 1 - 0 . 125 x 2 .
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Higher order equations To transform an arbitrary n th order equation y ( n ) = F ( t, y, y , · · · , y ( n - 1) ) (1) into a system of n first order equations, we extend the method in the last example by introducing the variables x 1 , x 2 , · · · , x n defined by x 1 = y, x 2 = y , · · · , x n = y ( n - 1) . (2) It then follows immediately that x 1 = x 2 , x 2 = x 3 , · · · , x n - 1 = x n , (3) and, from equation (1), x n = F ( t, x 1 , x 2 , · · · , x n ) . (4)
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Systems of first order equations Equations (3) and equation (4) form a system of n first order equations. The more general system has the following form
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