176_pdfsam_math 54 differential equation solutions odd

176_pdfsam_math 54 differential equation solutions odd -...

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Chapter 4 21. (a) With y ( t ) = e rt , y ( t ) = re rt , the equation becomes are rt + be rt = ( ar + b ) e rt = 0 . Since the function e rt is never zero on ( −∞ , ), to satisfy the above equation we must have ar + b = 0 . (b) Solving the characteristic equation, ar + b = 0, obtained in part (a), we get r = b/a . So y ( t ) = e rt = e bt/a , and a general solution is given by y = ce bt/a , where c is an arbitrary constant. 23. We form the characteristic equation, 5 r + 4 = 0, and find its root r = 4 / 5. Therefore, y ( t ) = ce 4 t/ 5 is a general solution to the given equation. 25. The characteristic equation, 6 r 13 = 0, has the root r = 13 / 6. Therefore, a general solution is given by w ( t ) = ce 13 t/ 6 . 27. Assuming that y 1 ( t ) = e t cos 2 t and y 2 ( t ) = e t sin 2 t are linearly dependent on (0 , 1), we conclude that, for some constant
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