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Unformatted text preview: McGill University Math 263: Differential Equations for Engineers CHAPTER 2: FIRST ORDER DIFFERENTIAL EQUATIONS In this lecture we will treat linear and separable first order ODE’s. 1 Linear Equation The general first order ODE has the form F ( x,y,y ) = 0 where y = y ( x ) . If it is linear it can be written in the form a ( x ) y + a 1 ( x ) y = b ( x ) where a ( x ) , a ( x ) , b ( x ) are continuous functions of x on some interval ( I ) . To bring it to normal form y = f ( x,y ) we have to divide both sides of the equation by a ( x ) . This is possible only for those x where a ( x ) 6 = 0 . After possibly shrinking I we as- sume that a ( x ) 6 = 0 on ( I ) . So our equation has the form (standard form) y + p ( x ) y = q ( x ) 0-0 with p ( x ) = a 1 ( x ) /a ( x ) , q ( x ) = b ( x ) /a ( x ) , both continuous on ( I ) . Solving for y we get the normal form for a linear first order ODE, namely y = q ( x )- p ( x ) y....
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