59X_errataSecondPrint.qxd /7/7 9:9 AM Page Second Printing Errata for Advanced Engineering Mathematics, by Dennis G. Zill and Michael R. Cullen (Yellow highlighting indicates corrected material) Page 7 The second equation in (5) is the result of using partial fractions on the left side of the first equation. Integrating and using the laws of logarithms gives ln y ln y x c y or ln y x c or y y exc. Page 56 Solution The equation is in standard form, and P(x) = and f (x) = x are continuous on ( q, q). The integrating factor is e dx = e x, and so integrating d dx 3ex y xe x Page 65 9. (t 3 y 5t y) dt + (t + 3y t) dy = Page 78 Example 5 Mixture of Two Salt Solutions Recall that the large tank considered in Section.3 held 3 gallons of a brine solution. Salt was entering and leaving the tank; a brine solution was being pumped into the tank at the rate of 3 gal/min, mixed with the solution there, and then the mixture was pumped out at the rate of 3 gal/min. The concentration of the salt in the inflow, or solution entering, was lb/gal, and so salt was entering the tank at the rate R in ( lb/gal) (3 gal/min) 6 lb/min and leaving the tank at the rate R out (x/3 lb/gal) (3 gal/min) x/ lb/min. From this data and (6) we get equation (8) of Section.3. Let us pose the question: If there were 5 lb of salt dissolved initially in the 3 gallons, how much salt is in the tank after a long time? Page 9 3. Leaking Conical Tank A tank in the form of a right-circular cone standing on end, vertex down, is leaking water through a circular hole in its bottom. (a) Suppose the tank is feet high and has radius 8 feet and the circular hole has radius inches. In Problem in Exercises.3 you were asked to show that the differential equation governing the height h of water leaking from a tank is dh dt 5 6h 3>. In this model, friction and contraction of the water at the hole were taken into account with c =.6, and g was taken to be 3 ft/s. See Figure.3. If the tank is initially full, how long will it take the tank to empty?
59X_errataSecondPrint.qxd /7/7 9:9 AM Page Page 9 Basic Definition is defined as a limit: If f (t) is defined for t, then the improper integral q Ks, t f t dt lim bsq b Ks, t f tdt. q Ks, t f t dt () Page Transforming (9) with respect to the variable x gives EI s Ys s 3 y s y sy y w L c L> s s s els> d or s Ys sy y w. EIL c L> s s s els> d Page 9 Similarly, + {t f (t)} + {t tf (t)} d {tf (t)} ds + d ds a d ds d +5 f t6b +5 f t6. ds Page 6 Example General Solution: Not an Integer By identifying and we can see from (9) that the general solution of the equation x y + xy + (x )y on (, ) is y c J / (x) + c J / (x). Page 8 Here the weights w i, i,,..., m are constants that generally satisfy w + w +... + w m and each k i, i,,..., m is the function f evaluated at a selected point (x, y) for which x n x x n +. We shall see that the k i are defined recursively. The number m is called the order of the method. Observe that by taking m, w and k f ( x n, y n ) we get the familiar Euler formula y n + y n + hf ( x n, y n ). Hence Euler s method is said to be a first-order Runge-Kutta method. Page 3 THEOREM 7. Criterion for Parallel Vectors Two nonzero vectors a and b are parallel if and only if a b.
59X_errataSecondPrint.qxd /7/7 9:9 AM Page 3 Page 3!! Solution The vectors P P and P P 3 can be taken as two sides of the triangle. Since!! P P i + j + 3k and P P 3 i 3j 5k, we have i j k! P! 3 3 P 3 33 i j 3 5 5 3 k P P 3 3 5 i + 8j 5k. Page 35 DEFINITION 8.5 Scalar Multiple of a Matrix If k is a real number, then the scalar multiple of a matrix A is ka ka p kan ka ka ka p kan ka ij mn. o o ka m ka m p kamn Page 379 are the same, it fol- Example Matrix with Two Identical Rows Since the second and third columns in the matrix A lows from Theorem 8.9 that 6 det A 3 3. 9 6 9 You should verify this by expanding the determinant by cofactors. Page 7 Multiplying these vectors, in turn, by the reciprocals of the norms K 3, K 6, and K 3, we obtain an orthonormal set 3 6,,. 3 6 3 6
59X_errataSecondPrint.qxd /7/7 9:9 AM Page Page 53 Subtracting (6) from (5) and using the fact that f/xy f/yx, we see that () becomes, after rearranging, R c a R Q y z b f P R a x z x b f Q a y x P bd da. y This last expression is the same as the right side of (3), which was to be shown. Page 535 Therefore, S curl F n ds To evaluate the latter surface integral, we use (5) of Section 9.3: xy x x ds S R x dx. S xy x da xy x x ds. xy x dy dx Bxy xyr dx (7) Page 5 Triple Integrals in Cylindrical Coordinates Recall from Section 9. that the area of a polar rectangle is A r * r, where r * is the average radius. From Figure 9.9(a) we see that the volume of a cylindrical wedge is simply V (area of base)(height) r * r z. Thus, if F(r,, z) is a continuous function over the region D, as shown in Figure 9.9(b), then the triple integral of F over D is given by f r, u Fr, u, z dv c f D R r, u Fr, u, z dz d da b a gu g u f r, u Fr, u, z r dz dr du. f r, u Page 595 Variation of Parameters Analogous to the procedure in Section 3.5, we ask whether it is possible to replace the matrix of constants C in () by a column matrix of functions u t u t Ut ± so that X o p tut u n t ()
59X_errataSecondPrint.qxd /7/7 9:9 AM Page 5 Page 65 Finally, when c 9, 9 3i. Thus the eigenvalues are conjugate complex numbers with negative real part. Figure.8(d) shows that solution curves spiral in toward the origin as t increases. Page 69 Solution If we let dx/dt y, then dy/dt x 3 x. From this we obtain the first-order differential equation which can be solved by separation of variables. Integrating dy dy>dt dx dx>dt x3 x, y y dy (x 3 x) dx gives y x x c. Page 659 By orthogonality we have and p cos mp p x dx, m 7, p cos mp p x sin np p p p p cos mp p x cos np p p x dx e, m n p, m n. x dx Finally, if we multiply () by sin(mx/p), integrate, and make use of the results and p sin mp p x dx, m 7, p sin mp p x sin np p p p p sin mp p x sin np p p x dx e, m n p, m n, x dx Page 67 Here T p so p. Since f is on the intervals (, ) and (, ), (8) be- Solution comes > > c n f xe inpx dx e inpx dx > einpx inp > np > e inp> e inp>. i >
59X_errataSecondPrint.qxd /7/7 9:9 AM Page 6 Page 7 In view of (6) and (7) of Section 3.8, the product solutions () can be written as u n x, t C n sin a npa () L t f nb sin np L x, where C n A n B n and n is defined by sin n A n /C n and cos n B n /C n. For n,, 3,... the standing waves are essentially the graphs of sin(nx/l), with a timevarying amplitude given by C n sin a npa L t f nb. Alternatively, we see from () that at a fixed value of x each product function u n (x, t) represents simple harmonic motion with amplitude C n sin(nx/l) and frequency f n na/l. In other words, each point on a standing wave vibrates with a different amplitude but with the same frequency. When n, u x, t C sin a pa L t f b sin p L x Page 76 Example Using the Cosine Transform The steady-state temperature in a semi-infinite plate is determined from u x u, x p, y 7 y u, y, up, y e y, y 7 u y, x p. y Page 768 Hence from (7), the Fourier coefficients are given by c f : ± c c c c 3 F ± i i ± i i f f f f 3.
59X_errataSecondPrint.qxd /7/7 9:9 AM Page 7 Page 8 Solution In this case, r and arg z /. From () with n, we obtain p> kp p> kp w k > c cos a b i sin a bd, k,,, 3. Thus, k, w > c cos p 6 i sin p d.696.7i 6 k, w > c cos 9p 6 k, w > c cos 7p 6 9p i sin d.7.696i 6 7p i sin d.696.7i 6 k 3, w 3 > c cos 5p 6 5p i sin d.7.696i. 6 Page 879 Example Order of Poles (a) Inspection of the rational function Fz z 5 z z 5z shows that the denominator has zeros of order at z and z 5, and a zero of order at z. Since the numerator is not zero at these points, it follows from Theorem 9. that F has simple poles at z and z 5, and a pole of order at z.