Advanced Mathematical Methods for Scientists and Engineers Episode 3 Part 9 doc

... the problem for u, (and hence the problem for y). As a check, then general solution for y is y = − 1 3 cos 2x + c 1 cos x + c 2 sin x. 1115 We guess a particular solution of the form y p = t e −t (a ... y 2 = e 3t . We compute the Wronskian of these solutions. W (t) =     e 2t e 3t 2 e 2t 3 e 3t     = e 5t We find a particular solution with variation of parameters. y p = − e...

Ngày tải lên: 06/08/2014, 01:21

... cos  2 √ a √ r  dr 1528 Hint 31 .17 Hint 31 .18 Hint 31 . 19 Hint 31 .20 Hint 31 .21 Hint 31 .22 Hint 31 . 23 1506 Exercise 31 .12 Find the inverse Laplace transform of ˆ f(s) = 1 s 3 − 2s 2 + s − 2 with the following methods. 1. ... ı)  s=−ı = ı 1 4 1 530 We substitute this into the third equation. sˆy 3 = −ˆy 3 − ˆy 3 s − ˆy 3 s 2 + 6 s 4 (s 3 + s 2 + s + 1)ˆy 3...

Ngày tải lên: 06/08/2014, 01:21

... 7 .32 Hint 7 .33 Hint 7 .34 Hint 7 .35 Hint 7 .36 Hint 7 .37 30 0 Now to define the branch. We make a choice of angles. z + 3 = r 1 e ıθ 1 , −π < θ 1 < π z = r 2 e ıθ 2 , − π 2 < θ 2 < 3 2 z ... z 1 /3 = 3 √ z e ık2π /3 , k = 0, 1, 2 2. g(w) = w 3 = |w| 3 e 3 arg(w) 32 8 3 1+ı = e (1+ı) log 3 = e (1+ı)(ln 3+ ı2πn) = e ln 3 2πn e ı(ln 3+ 2πn) , n ∈ Z  √...

Ngày tải lên: 06/08/2014, 01:21

... =  3 − √ 2  1 /3  3 + √ 2  1 /3 ( 3 + 2) 1 /3 = 3  3 + √ 2 e ıπ /3 3  3 − √ 2 e ıπ /3 3 √ 1 e ıπ /3 = 3 √ 7 e ıπ = − 3 √ 7 The value of the function is w = 3 √ abc e ı(α+β+γ) /3 . Consider the ... real variable counterpart. 36 4 With f(z) = z 1 /3 (z −ı) −1 /3 (z + ı) −1 /3 = 3 √ r e ıθ /3 1 3 √ s e −ıφ /3 1 3 √ t e −ıψ /3 = 3  r st e ı(θ−φ−ψ) /3...

Ngày tải lên: 06/08/2014, 01:21

... for the coefficients. a + b + c = 2 4a + 2b + c = 6 9a + 3b + c = 12 p(n) = n 2 + n We e xamine the first few partial sums. S 1 = 1 2 S 2 = 2 3 S 3 = 3 4 S 4 = 4 5 592 5. ∞  n=1 ln (2 n ) ln (3 n ) ... 12. 13 CONTINUE. 1. ∞  n=0 z n (z + 3) n 2. ∞  n=2 Log z ln n 3. ∞  n=1 z n 4. ∞  n=1 (z + 2) 2 n 2 594 Thus the series converges absolutely for |z| < 1/4. 6. By the Ca...

Ngày tải lên: 06/08/2014, 01:21

... integral converges for (a) < 1. 725 Figure 13. 8: The real and imaginary part of the integrand for several values of α. Note that the integral exists for all nonzero real α and that lim α→0 +  1 −1 1 x ... = ı  = ıπ 3 lim z→ı 1 z + ı = π 3 2 Now we return to Equation 13. 1. 2  ∞ 0 ln 2 x 1 + x 2 dx + ı2π  ∞ 0 ln x 1 + x 2 dx = π 3 4 We equate the real and imaginary...

Ngày tải lên: 06/08/2014, 01:21

... x n F  1, y x  . (Just formally substitute 1/x for λ.) For example, xy 2 , x 2 y + 2y 3 x + y , x cos(y/x) are homogeneous functions of orders 3, 2 and 1, respectively. Euler’s theorem for a homogeneous ... αy(t). 775 1 2 3 4 4 8 12 16 1 2 3 4 4 8 12 16 Figure 14.1: The p opulation of bacteria. 1 2 3 4 32 64 96 128 1 2 3 4 32 64 96 128 Figure 14.2: The discrete po...

Ngày tải lên: 06/08/2014, 01:21

... equation dw dz + 1 z 2 + 9 w = 0. We factor the denominator of the fraction to see that z = 3 and z = − 3 are regular singular points. dw dz + 1 (z − 3) (z + 3) w = 0 We make the transformation z = 1/ζ ... rearranging terms to form exact derivatives. 4yy  − xy  − y + 1 −9x 2 = 0 d dx  2y 2 − xy  + 1 − 9x 2 = 0 2y 2 − xy + x −3x 3 + c = 0 y = 1 4  x ±  x 2 − 8(c + x − 3x 3 ) ...

Ngày tải lên: 06/08/2014, 01:21

... of η to be zero.   4 3 −2 8 −6 −4 −4 3 2     0 0 η 3   = c 1   1 0 2   + c 2   0 2 3   −2η 3 = c 1 , −4η 3 = 2c 2 , 2η 3 = 2c 1 − 3c 2 c 1 = c 2 , η 3 = − c 1 2 888 We see that ... 0     e 2 e 2 e 2 /2 0 e 2 e 2 0 0 e 2     0 3 −2 0 −1 −1 1 −1 −1   e A =   0 2 2 3 1 −1 −5 3 5   e 2 2 8 59 Thus there are two eigenvectors.   4 3 −2 8 −6 −4 −...

Ngày tải lên: 06/08/2014, 01:21

... =     e x e 2x e x 2 e 2x     = 2 e 3x − e 3x = e 3x . 91 0 Consider L[y] = p n y (n) + ···+ p 0 y. If each of the p k is k times continuously differentiable and u and v are n times continuously differentiable ... 0 ¯y (n) + p n−1 ¯y (n−1) + ··· + p 0 ¯y = 0 L [¯y] = 0 9 03 16 .3 Transformation to a First Order System Any linear differential equation can be put in the form of a...

Ngày tải lên: 06/08/2014, 01:21