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Lesson 6a - Bernoulli Differential Equations

U2 2 u 1 ca xu 1 cb x 1 1 u 4 xu 2 x 2 2 u

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Unformatted text preview: P y '4 xy 2 x y y ' a ( x) y b( x) y c 1 a ( x ) 4 x b( x ) 2 x c yu 1 1 c y u2 2 u ' (1 c)a ( x)u (1 c)b( x) 1 1 u '( )4 xu (2 x) 2 2 u '2 xu x 2 xdx I ( x) e x 2 I ( x) e I ( x)b( x)dx e x2 xdx Let v x 2 , then dv 2 xdx y ' 2 x y 2 xy, y 1 e v dv 2 1 ev c 2 1 x2 e c 2 1 u I ( x)b( x)dx I ( x) 1 1 x2 u x 2 ( e c) e2 1 x2 u ce 2 1 x2 y ce 2 1 x2 2 y ( ce ) 2 y ( ln(2) ) 9 Example 2 Continued Determine the general solution to the IVP 1 x2 2 y ( ce ) 2 1 9 ( ce ( 2 1 12 9 ( c ln 2 ) 2 e 1 1 9 ( c )2 2 2 1 1 3( c ) 2 2 6 1 c ln( 2 ) ) 2 2 1 9 ( ce ln 2 ) 2 2 y ' 2 x y 2 xy, y ) 5c 1 x2 2 y ( 5e ) 2 y ( ln(2) ) 9...
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