Advanced Engineering Math 9th Edition with Mathematica by Erwin Kreyszig

By Erwin Kreyszig

Booklet via Kreyszig, Erwin

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14. 5. 2. 5. 8. 16. Team Project. (A) The student should realize that the present steps are the same as in the general derivation of the method in Sec. 1. An advantage of such specific derivations may be that the student gets a somewhat better understanding of the method and feels more comfortable with it. Of course, once a general formula is available, there is no objection to applying it to specific cases, but often a direct derivation may be simpler. In that respect the present situation resembles, for instance, that of the integral solution formula for first-order linear ODEs in Sec.

CAS Experiment. The choice of ␻ needs experimentation, inspection of the curves obtained, and then changes on a trial-and-error basis. It is interesting to see how in the case of beats the period gets increasingly longer and the maximum amplitude gets increasingly larger as ␻ /(2␲) approaches the resonance frequency. 26. If 0 Ϲ t Ϲ ␲, then a particular solution yp ϭ K0 ϩ K1t ϩ K2 t 2 gives ypЉ ϭ 2K2 and 1 ypЉ ϩ yp ϭ K0 ϩ 2K2 ϩ K1t ϩ K2 t 2 ϭ 1 Ϫ ᎏ2 t 2; ␲ thus, 1 K2 ϭ Ϫ ᎏ2 , ␲ K1 ϭ 0, 2 K0 ϭ 1 Ϫ 2K2 ϭ 1 ϩ ᎏ2 .

14. For the second given function the answer is 40e 5x and for the third it is Ϫ5x 2 ϩ 8x ϩ 2. 4x) 4(D ϩ _12␲I)2, y ϭ (c1 ϩ c2 x)e؊␲ x/2 y is a solution, as follows from the superposition principle in Sec. 1 because the ODE is homogeneous linear. In the application of l’Hôpital’s rule, y is regarded as a function of ␮, the variable that is approaching the limit, whereas ␭ is fixed. qxd 9/21/05 38 10:57 AM Page 38 Instructor’s Manual Accordingly, differentiation of the numerator with respect to ␮ gives xe␮x Ϫ 0, and differentiation of the denominator gives 1.

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