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Lecture 9

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University of Ottawa
Leslie Shiell

ECO 3145 Mathematical Economics I Lecture 9 FirstOrder Differential EquationsChiang ch 151 153 156 157SummaryI Introduction II Solution method integration III Dynamic characteristics of a solution IV General solution method V Qualitative analysisphase diagramsI Introduction1 time pathcontinuoustime analogue of a sequencethe value of a variable expressed as a function of time t t2 notationor more precisely tx txt t1 2 differential equation DE definitionvariables measured in continuous timean equation which specifies the instantaneous change in the value of a variableat least one term of the equation is a derivative or a differential linear form t dx or equivalently 1 tb txtb tx txdtnonlinear formexample Solow growth model definitions of variables and parameters in previous lecture tk ti tk tk tk ftk t Ak 31459doc Shiell 13 solutiontx an expression for the time path of the dependent variableas a function of tb the time paths of the independent variables ie timea given value of the dependent variable such as the initial value x0 II Solution method integration1 backward solution rewrite 1 in the following formtb tx tx t trick multiply both sides of the equation bye tttb e tx tx e dt observe that the lefthand side of this equation is equivalent totx edt make this substitution into the equation yieldingdtt tb e tx edt 0 tT t integrate both sides of the equation fromtowhere T denotes a particular value of t for which you want to know the solutionTTdttdt tb e dt tx e dt00simplify Tt0Tdt tb e 0x e Tx e0TtTdt tb e 0x Tx e 0TtTdt tb e 0x Tx e031459doc Shiell 2
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