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FIRST ORDER LINEAR DIFFERENTIAL EQUATIONS - Copy (2)

FIRST ORDER LINEAR DIFFERENTIAL EQUATIONS - Copy (2) - A...

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A. Alaca MATH 1005F Fall 2008 1 FIRST ORDER LINEAR DIFFERENTIAL EQUATIONS A frst order linear differential equation is an equation oF the Form y ± + P ( x ) y = Q ( x )( * ) where P ( x ) and Q ( x ) continuous Functions on a given interval. Method of solution: We are looking For an integrating Factor I ( x ) such that when we multiply both sides oF the equation (*) by I ( x ), leFt hand side oF the equation would be d dx ( Iy ). I ( x ) y ± + P ( x ) yI ( x )= Q ( x ) I ( x ) d dx ( QI = ± Q ( x ) I ( x ) dx y = 1 I ( x ) ± Q ( x ) I ( x ) dx . How to ±nd I ( x ) : I ( x )( y ± + P ( x ) y d dx ( I ( x ) y ) . I ( x ) y ± + P ( x ) I ( x ) y = I ± ( x ) y + I ( x ) y ± P ( x ) I ( x I ± ( x dI dx P ( x ) dx = dI I ln | I | = ± P ( x ) dx e ln | I | = e ² P ( x ) dx | I | = e ² P ( x ) dx Since we do not need most general integrating Factor, we take
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