Chapter 2 Lecture notes

Chapter 2 Lecture notes - McGill University Math 263...

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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 ) = 0 where y = y ( x ) . If it is linear it can be written in the form a 0 ( x ) y 0 + a 1 ( x ) y = b ( x ) where a 0 ( x ) , a ( x ) , b ( x ) are continuous functions of x on some interval ( I ) . To bring it to normal form y 0 = f ( x, y ) we have to divide both sides of the equation by a 0 ( x ) . This is possible only for those x where a 0 ( x ) 6 = 0 . After possibly shrinking I we as- sume that a 0 ( x ) 6 = 0 on ( I ) . So our equation has the form (standard form) y 0 + p ( x ) y = q ( x ) 0-0
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with p ( x ) = a 1 ( x ) /a 0 ( x ) , q ( x ) = b ( x ) /a 0 ( x ) , both continuous on ( I ) . Solving for y 0 we get the normal form for a linear first order ODE, namely y 0 = q ( x ) - p ( x ) y. 1.1 Linear homogeneous equation Let us first consider the simple case: q ( x ) = 0 , namely, d y d x + p ( x ) y = 0 . With the chain law of derivative, one may write y 0 ( x ) y = d d x ln £ y ( x ) / = - p ( x ) , integrating both sides, we derive ln y ( x ) = - Z p ( x )d x + C,
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