Lesson 4a - Homogeneous

Substituteandifallofyouryvariablesdisappear x

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Unformatted text preview: omogenous strategy outlined below: y u u xu ' y ' 1. Substitute and if all of your y variables disappear x (and you only have the x in front of the u’ in your equation), then this will be a separable equation most definitely. 2. Solve the separable equation in terms of u, u’, and x. y u and solve for y (if possible). 3. Substitute back x Example 1 Determine the general solution to y x xe y y' x x y y y' e x x y Let u x Th Thus y ' xu 'u e u y x xy ' xe y du 1 dx dx dx x 1 e du x dx u e u ln | x | c e u c ln | x | xu 'u eu u xu ' e u e u u ' 1 x ln(e u ) ln(c ln | x |) u ln(c ln | x |) u ln(c ln | x |) y ln(c ln | x |) x y x ln(c ln | x |) Example 2 x 2 y xy 2 y 3 y' Find the general solution to the following ODE: 2 2 3 xy 2 x y xy y 1 u du 1 y' 2 2 2 xy xy xy u 1 u 0 u 1 dx dx xdx x y u 1 y' 1 1 u y x R 1 1 u 1 du xdx y x 1 u 1 Let u , (note u ) 1 1 1 y x 1 du dx u 1 u 1 u 1 Thus y ' xu 'u x xu 'u u 1 u 1 1 xu ' 1 u u 1 xu ' u u 1 u' u 1 x y x u ln | u 1 | ln | x | c e u ln|u 1| e ln| x| c eu e ln| x|e c e ln|u 1| eu Ax u 1 e Ax y 1 x y x y e Ax( 1) x Example 3 xy y x2 y2 x y y x y' y 1 ( )2 x x Determine the general solution to the following ODE: y ' xy xy x2 x2 y2 x2 y2 x2 y xy x x y 1 ( y )2 x y Thus y ' xu 'u Let u , x 2 u u 2 1 u u xu ' 1 u2 xu 'u 1 u 1 u' u x 2 1u 1 ( )u ' uu x 1 1 ( u )u ' u x 1 du 1 ( u u ) dx dx x dx 1 1 ( u )du dx u x u2 ln | u | ln | x | c 2 2 ln | u | u 2 2 ln | x | c y y2 2 ln | | 2 ln | x | ( ) c x x y y2 2 ln | 2 | ( ) c x x...
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This note was uploaded on 02/07/2014 for the course MATH 1004 taught by Professor Mark during the Summer '00 term at Carleton CA.

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