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# L10 - Examples Example Consider the IVP 2y 3xy 3y = 5x 2x...

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Examples Example: Consider the IVP 2 x 2 y 00 + 3 xy 0 - 3 y = 5 x y (1) = 1 , y 0 (1) = 1 . ( NH ) Strategy: find a FS for the associated ho- mogeneous SOLODE 2 x 2 y 00 + 3 xy 0 - 3 y = 0 , ( H ) then, using Variation of Parameters, find a particular solution Y ( x ) of ( NH ) , which will give us the general solution y ( x ) = Y ( x ) + c 1 y 1 ( x ) + c 2 y 2 ( x ) , ( G ) then accommodate the Initial Conditions. See that ( H ) has the solution y 1 ( x ) = x .

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Panic! I have forgotten how Reduction of Order works. Actually, I do remember that it uses Abel’s theorem: there is a second solution y 2 such that W [ y 1 , y 2 ]( x ) = e - R p ( x ) dx . Now y 1 y 0 2 - y 2 y 0 1 = e - R 3 / 2 x dx = x - 3 / 2 = x - 3 / 2 = y 1 y 0 2 - y 2 y 0 1 y 2 1 = x - 3 / 2 x 2 = x - 7 / 2 = y 2 y 1 0 = x - 7 / 2 = y 2 y 1 = Z x - 7 / 2 dx = - 2 5 x - 5 / 2 2
= y 2 ( x ) = - 5 2 x - 3 / 2 Thus { x, x - 3 / 2 } is a FS for ( H ) . It has Wronskian W [ y 1 , y 2 ] = - 5 2 x - 3 / 2 . Now find a particular solution Y of ( NH ) . Panic! I have forgotten how Variation of Parameters works. Actually, I remember that it means looking for a particular so- lution of the form Y ( x ) = v 1 ( x ) y 1 ( x ) + v 2 ( x ) y 2 ( x ) with variable parameters v 1 , v 2 . So I take some scratch paper and derive it right quick: 3

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Y 00 = v 1 y 00 1 + v 2 y 00 2 + v 0 1 y 0 1 + v 0 2 y 0 2 p Y 0 = v 1 p y 0 1 + v 2 p y 0 2 + p [ v 0 1 y 1 + v 0 2 y 2 | {z } =0 ] q Y = v 1 q y 1 + v 2 q y 2 L [ Y ] = v 1 L [ y 1 ] + v 2 L [ y 2 ] | {z } =0 + v 0 1 y 0 1 + v 0 2 y 0 2 | {z } = g results in the two linear equations v 0 1 y 1 + v 0 2 y 2 = 0 , v 0 1 y 0 1 + v 0 2 y 0 2 = g .
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