244_pdfsam_math 54 differential equation solutions odd

244_pdfsam_math 54 differential equation solutions odd -...

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Chapter 4 By substituting π + arctan ( 3 / 2 ) for φ in the last equation above and by requiring that t be greater than zero, we obtain t n = ( π/ 3) + ( n 1) π arctan ( 3 / 2 ) 2 3 ,n =1 , 2 , 3 ,... . We see that the solution curve given by equation (4.13) above will touch the exponential curves y ( t )= ± ± p 7 / 12 ² e 2 t when we have r 7 12 e 2 t sin ± 2 3 t + φ ² = ± r 7 12 e 2 t , where φ = π + arctan ( 3 / 2 ) . This will occur when sin ( 2 3 t + φ ) = ± 1. Since sin θ = ± 1 when θ =( 2) + for any integer m , we see that the times T m , when the solution touches the exponential curves, satisfy 2 3 T m + φ = π 2 + T m = ( 2) + φ 2 3 , where φ = π + arctan ( 3 / 2 ) and m is an integer. Again requiring that t be positive we see that y ( t ) touches the exponential curve when T m = ( 2) + ( m 1) π
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