02_ode_1st_2nd_order_fromHutchinson

02_ode_1st_2nd_order_fromHutchinson - AM105b, Spring 2005...

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AM105b, Spring 2005 FIRST AND SECOND ORDER ODEs—ESSENTIALS General solution to linear first order ode : ( ) () y ax y hx += 1 ( ) ( ) P y x c y x yx =+ with homogeneous solution 0 1 x x ad y xe xx - = and particular solution 1 1 1 ( ) x P x h y x y xd y x x x = where c , 0 x 1 x are constants that can be chosen at the discretion of the analyst. General solution to constant coefficient linear second order ode : y a y b y hx ′′ ++= 1 1 22 ( ) ( ) ( ) P y x cy x c yx yx =++ Homogeneous solutions —three possibilities, depending on a b If 2 /4 ab , 12 , yeye mm == / 2 / 4 , / 2 a a b aab = - +- = - -- (Note that this includes the case for 0 b = , with 1, ax y ye - ) If 2 < , si n , cos y e x y ex nn 2 /2 , a ba mn = - =- 0 a = si n , cos y x yx 2 = , the method of reduction of order (see below) can be used to show , y e y xe a m 0 y Particular solution —a general representation obtained by reduction of order. Let H denote any homogeneous solution (i.e. 1 1 ( ) ( ) H x cyx , for some choice of 1 c 2 c .). By the method of reduction of order (see below): 0 2 ( ) ()() ax x a PH x H e y x e y hd x xxx - = 1 ( ) x PP x y x yd = 0 x 1 x can be chosen at the discretion of the analyst.
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02_ode_1st_2nd_order_fromHutchinson - AM105b, Spring 2005...

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