Advanced Mathematical Methods for Scientists and Engineers Episode 3 Part 4 pot

... exp  −  p n−1 (x) dx  . Example 16 .4. 4 Consider the differential equation y  − 3y  + 2y = 0. The Wronskian of the two independent solutions is W (x) = c exp  −  3 dx  = c e 3x . For the choice of solutions ... that e Jt =   e 2t t e 2t 0 0 e 2t 0 0 0 e 3t   8 94 Particular Solutions. Any function, y p , that satisfies the inhomogeneous equation, L[y p ] = f (x), is called a...

Ngày tải lên: 06/08/2014, 01:21

... 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 population of bacteria and a continuous population approximation. That is, ... doubles every hour. For the continuous problem, we assume that this i s true for y(t). We write this as an equation: y  (t) =...

Ngày tải lên: 06/08/2014, 01:21

... 9x 2 = 0 2y 2 − xy + x −3x 3 + c = 0 y = 1 4  x ±  x 2 − 8(c + x − 3x 3 )  2. We consider the differential equation, (2x − 2y)y  + (2x + 4y) = 0. 837 14. 11 Solutions Solution 14. 1 1. y  (x) y(x) = ... equation u  − sin(1/ζ) ζ u = 0. 840 -5 -4 -3 -2 -1 1 -3 -2 -1 1 Figure 14. 11: The function 2(1 − x) e x −1. Solution 14. 11 1. We consider the differential equation, (4y −x...

Ngày tải lên: 06/08/2014, 01:21

... for r > 0, it must be a global minimum. The cup has a radius of 4 3 √ π cm and a height of 4 3 √ π . 106 a x x x x x x∆ 1 2 3 i n-2 n-1 b f( ) ξ 1 Figure 4. 2: Divide -and- Conquer Strategy for ... f  (r), 2πr − 128 r 2 = 0, 2πr 3 − 128 = 0, r = 4 3 √ π . The second derivative of the surface area is f  (r) = 2π + 256 r 3 . Since f  ( 4 3 √ π ) = 6π, r = 4...

Ngày tải lên: 06/08/2014, 01:21

... Integral Formula to evaluate the integrals along C 1 and C 2 .  C (z 3 + z + ı) sin z z 4 + ız 3 dz =  C 1 (z 3 + z + ı) sin z z 3 (z + ı) dz +  C 2 (z 3 + z + ı) sin z z 3 (z + ı) dz = ı2π  (z 3 + ... integrand to s ee that there are singularities at the cube roots of 9. z z 3 − 9 = z  z − 3 √ 9  z − 3 √ 9 e ı2π /3  z − 3 √ 9 e −ı2π /3  Let C 1 , C 2 an...

Ngày tải lên: 06/08/2014, 01:21

... the 656 Hint 13. 39 Hint 13. 40 Definite Integrals Involving Sine and Cosine Hint 13. 41 Hint 13. 42 Hint 13. 43 Hint 13. 44 Make the changes of variables x = sin ξ and then z = e ıξ . Infinite Sums Hint 13. 45 Use ... ıa)     z=a = 1 4a 3 Res  1 z 4 − a 4 , z = −a  = 1 (z − a)(z − ıa)(z + ıa)     z=−a = − 1 4a 3 Res  1 z 4 − a 4 , z = ıa  = 1 (z − a)(z +...

Ngày tải lên: 06/08/2014, 01:21

... components 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 , 4 3 = 2c 2 , 2η 3 = 2c 1 − 3c 2 c 1 = c 2 , η 3 = − c 1 2 888 We see ... 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 2 859 Thus there are two eigenvectors.   4 3 −2 8 −...

Ngày tải lên: 06/08/2014, 01:21

... make this an exact equation. d dx  e x 3 /3 y  = c 1 e x 3 /3 e x 3 /3 y = c 1  e x 3 /3 dx + c 2 y = c 1 e −x 3 /3  e x 3 /3 dx + c 2 e −x 3 /3 945 Exercise 17.21 Find the general solution ... real and positive. 944 Method 1. Note that this is an exact equation. d dx (y  − x 2 y) = 0 y  − x 2 y = c 1 d dx  e −x 3 /3 y  = c 1 e −x 3 /3 y = c 1 e x 3 /...

Ngày tải lên: 06/08/2014, 01:21

... x  = dx dy . dy dx = 1 y 3 − xy 2 dx dy = y 3 − xy 2 x  + y 2 x = y 3 Now we have a first order equation for x. d dy  e y 3 /3 x  = y 3 e y 3 /3 x = e −y 3 /3  y 3 e y 3 /3 dy + c e −y 3 /3 Example 18 .3. 2 Consider ... − 1 4 y 4 + c 1 u =  2c 1 − 1 2 y 4  1/2 y  =  2c 1 − 1 2 y 4  1/2 dy (2c 1 − 1 2 y 4 ) 1/2 = dx Integrating gives us the implici...

Ngày tải lên: 06/08/2014, 01:21

... y 1/2 exp  − 2y 3/ 2 3  x exp  − 2y 3/ 2 3  = −exp  − 2y 3/ 2 3  + c 1 x = −1 + c 1 exp  2y 3/ 2 3  x + 1 c 1 = exp  2y 3/ 2 3  log  x + 1 c 1  = 2 3 y 3/ 2 y =  3 2 log  x + 1 c 1  2 /3 y =  c + 3 2 log(x ... 3p  )u  exp  − 1 3  p(x) dx  y  =  u  − pu  + 1 3 (p 2 − 3p  )u  + 1 27 (9p  − 9p  − p 3 )u  exp  − 1 3  p(x) dx  y...

Ngày tải lên: 06/08/2014, 01:21