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m2k_dfq_inford

# m2k_dfq_inford - First Order Differential Equations an...

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Unformatted text preview: First Order Differential Equations; an Introduction Differential Equations A differential equation has much in com- mon with a parametric equation , as discussed in the section on that topic. We recall that such an equation has the form f ( x, y ) = 0 and, in general, the solution set consists of one or more functions y = y ( x ). An n- th order differential equation has, or can be put in, the form F x, y, dy dx , d 2 y dx 2 , ..., d n y dx n ≡ F x, y, y , y 00 , ..., y ( n ) = 0 . Since it involves the two variables x and y , we expect to find solutions in the form y = y ( x ), as in the case of parametric equations – which can be thought of as differential equations of order zero. However, since the differential equation also involves certain derivatives of y ( x ), we are looking for functions y ( x ) possessing derivatives of the orders indicated in the differential equation and such that substitution into the differential equation yields an identity, i.e., F x, y ( x ) , y ( x ) , y 00 ( x ) , ..., y ( n ) ( x ) ≡ for an appropriate range of values of the independent variable. Example 1 The second order differential equation d 2 y dx 2 + 9 y = 0 has any of the functions y ( x ) = a cos 3 x + b sin 3 x as a solution, where a and b may be any constants (i.e., any constant functions of x ). Thus cos 3 x, sin 3 x, 2 cos 3 x- 5 sin 3 x , etc., are all solutions (corresponding to a = 1 , b = 0 , a = 0 , b = 1 and a = 2 , b =- 5, respectively. To check this we simply observe that d 2 dx 2 ( a cos 3 x + b sin 3 x ) = a d 2 cos 3 x dx 2 + b d 2 sin 3 x dx 2 1 = a (- 9 cos 3 x ) + b (- 9 sin 3 x ) =- 9 ( a cos 3 x + b sin 3 x ) . First Order Differential Equations A first order differential equa- tion (for the function y ( x )) is an equation generally involving x, y ( x ) and dy dx : F x, y, dy dx ! = 0 . We ordinarily assume that this equation can be solved for dy dx to give the equation in its standard form dy dx = f ( x, y ) . We commonly assume that the function f ( x, y ) is defined in a rectan- gular region a ≤ x ≤ b ; a 1 ≤ y ≤ b 1 ; either or both of these intervals could be replaced by an infinite interval such as (-∞ , ∞ ); we often do not specify these intervals explicitly. The simplest definition of a solution y ( x ) is a function defined on at least part of [ a , b ], with derivative y ( x ) such that, wherever y ( x ) and f ( x, y ( x )) are defined, y ( x ) ≡ f ( x, y ( x )) . Example 2 The right hand side f ( x, y ) = y 2 of the differential equation dy dx = y 2 is defined for-∞ < x < ∞ ,-∞ < y < ∞ and has y ( x ) = 1 1- x as a solution on the intervals (-∞ , 1) and (1 , ∞ ). There are some exceptional cases that require a relaxed definition of what we mean by a solution....
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m2k_dfq_inford - First Order Differential Equations an...

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