178_pdfsam_math 54 differential equation solutions odd

178_pdfsam_math 54 differential equation solutions odd -...

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Chapter 4 For t =0 , y 0 1 (0) = 3 · 0 2 =0and y 0 2 (0) = 0. The latter follows from the fact that one-sided derivatives of y 2 ( t ), 3 t 2 and 3 t 2 , are both zero at t =0 . A lso , y 1 (0) = y 2 (0) = 0. Hence W [ y 1 ,y 2 ](0) = ± ± ± ± ± 00 00 ± ± ± ± ± =0 , and so W [ y 1 ,y 2 ]( t ) 0on( −∞ , ). This result does not contradict part (b) in Prob- lem 34 because these functions are not a pair of solutions to a homogeneous linear equation with constant coefficients. 37. If y 1 ( t )and y 2 ( t ) are solutions to the equation ay 0 + by 0 + c = 0, then, by Abel’s formula, W [ y 1 ,y 2 ]( t )= Ce bt/a ,where C is a constant depending on y 1 and y 2 .Thu s ,i f C 6 =0 ,then W [ y 1 ,y 2 ]( t ) 6 =0forany t in ( −∞ , ), because the exponential function, e bt/a , is never zero. For C =0 , W [ y 1 ,y 2 ]( t ) 0on( −∞ , ). 39. (a) A linear combination of y 1 ( t )=1, y 2 ( t )= t ,and y 3 ( t )= t 2 , C 1 · 1+ C 2 · t + C 3 · t 2 = C 1 + C 2 t + C
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This note was uploaded on 03/29/2010 for the course MATH 54257 taught by Professor Hald,oh during the Spring '10 term at University of California, Berkeley.

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