lecture_23 (dragged)

lecture_23 (dragged) - 2 ( t ) ° = 0 . Then L [ y ] = 0 if...

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MA 36600 LECTURE NOTES: WEDNESDAY, MARCH 11 Higher Order Linear Equations Review. Recall that a second order linear equation is in the form a ( t ) y °° + b ( t ) y ° + c ( t ) y = f ( t ) . We found that the general solution is in the form y ( t )= c 1 y 1 ( t )+ c 2 y 2 ( t )+ Y ( t ) where { y 1 ( t ) ,y 2 ( t ) } is a fundamental set of solutions to the homogeneous equation a ( t ) y °° + b ( t ) y ° + c ( t ) y =0 and Y = Y ( t ) is a particular solution to the nonhomogeneous equation a ( t ) Y °° + b ( t ) Y ° + c ( t ) Y = f ( t ) . We give a diFerent way to view this. De±ne the following operator: L [ y ]= a ( t ) d 2 y dt 2 + b ( t ) dy dt + c ( t ) y. This is a linear operator , i.e., given functions f = f ( t ) and g = g ( t ) as well as constants c 1 and c 2 ,wehave L ° c 1 f + c 2 y ± = a ( t ) d 2 dt 2 ² c 1 f ( t )+ c 2 g ( t ) ³ + b ( t ) d dt ² c 1 f ( t )+ c 2 g ( t ) ³ + c ( t ) ² c 1 f ( t )+ c 2 g ( t ) ³ = a ( t ) ² c 1 d 2 f dt 2 + c 2 d 2 g dt 2 ³ + b ( t ) ² c 1 df dt + c 2 dg dt ³ + c ( t ) ² c 1 f + c 2 g ³ = c 1 ² a ( t ) d 2 f dt 2 + b ( t ) df dt + c ( t ) f ³ + c 2 ² a ( t ) d 2 g dt 2 + b ( t ) dg dt + c ( t ) g ³ = c 1 L [ f ]+ c 2 L [ g ] . We wish to ±nd all functions y = y ( t ) such that L [ y ]= f ( t ). In a sense, we would like to compute y ( t )= L 1 [ f ]. We do this in two steps: ²irst we ±nd functions y 1 = y 1 ( t ) and y 2 = y 2 ( t ) such that L [ y 1 ]= L [ y 2 ] = 0 and W ´ y 1 ,y 2 µ ( t )= y 1 ( t ) y 2 ( t ) y ° 1 ( t ) y ° 2 ( t ) = y 1 ( t ) y ° 2 ( t ) y ° 1 ( t ) y
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Unformatted text preview: 2 ( t ) ° = 0 . Then L [ y ] = 0 if and only if y ( t ) = c 1 y 1 ( t ) + c 2 y 2 ( t ) . Similarly, L [ y ] = f ( t ) if and only if y ( t ) = · c 1 y 1 ( t ) + c 2 y 2 ( t ) ¸ + ¹ t f ( τ ) a ( τ ) y 1 ( τ ) y 2 ( t ) − y 1 ( t ) y 2 ( τ ) y 1 ( τ ) y ° 2 ( τ ) − y ° 1 ( τ ) y 2 ( τ ) dτ. General Theory. Recall that we use the notation y ( k ) = d k y dt k to denote the n th derivative. We say that an equation of the form F º t, y, y (1) , . . . , y ( n ) » = 0 is an n th order diferential equation . In particular, we can express the highest order derivative in terms of the lower order derivatives: d n y dt n = G º t, y, y (1) , . . . , y ( n − 1) » . 1...
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This note was uploaded on 11/30/2011 for the course MATH 366 taught by Professor Edraygoins during the Spring '09 term at Purdue.

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