Advanced Mathematical Methods for Scientists and Engineers Episode 3 Part 7 pps

... 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 ...

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 ... equation of order n − 1 for u  . Writing the derivatives of e u(x) , d dx e u = u  e u d 2 dx 2 e u = (u  + (u  ) 2 ) e u d 3 dx 3 e...

Ngày tải lên: 06/08/2014, 01:21

... 1) 1 3 x 3 + x  1 3 − 1 2 − 1 3 x 3 + 1 2 x 2  y = 1 6 (x 3 − x). Example 21 .7. 4 Find the solution to the differential equation y  − y = sin x, that is bounded for all x. The Green function for ... the homogeneous solutions to y p and it will still be a particular solution. For example, η p = − 1 3 sin(2x) − 1 3 sin x = − 2 3 sin  3x 2  cos  x 2  is a partic...

Ngày tải lên: 06/08/2014, 01:21

... ı  2 n by successive squaring.  √ 3 + ı  2 = 2 + ı2 √ 3  √ 3 + ı  4 = −8 + ı8 √ 3  √ 3 + ı  8 = −128 − ı128 √ 3  √ 3 + ı  16 = 32 76 8 + 32 76 8 √ 3 Next we multiply  √ 3 + ı  4 and  √ 3 + ı  16 to obtain ... answer.  √ 3 + ı  20 =  32 76 8 + 32 76 8 √ 3  −8 + ı8 √ 3  = −524288 − ı524288 √ 3 Since we know that arctan  √ 3, 1  = π/6, it...

Ngày tải lên: 06/08/2014, 01:21

... ı2 √ 3  −2 + ı2 √ 3  4  −1 =   −2 + ı2 √ 3  −8 − ı8 √ 3  2  −1 =  −2 + ı2 √ 3  −128 + ı128 √ 3  −1 =  −512 − ı512 √ 3  −1 = 1 512 −1 1 + ı √ 3 = 1 512 −1 1 + ı √ 3 1 − ı √ 3 1 ... u 2 3 v =  (u 0 w 0 + u 1 w 1 + u 2 w 2 + u 3 w 3 ) + ı (−u 1 w 0 + u 0 w 1 + u 3 w 2 − u 2 w 3 ) +  (−u 2 w 0 − u 3 w 1 + u 0 w 2 + u 1 w 3 ) + k (−u 3 w 0 +...

Ngày tải lên: 06/08/2014, 01:21

... ∆r cos θ and y = ∆r sin θ. f  (0) = lim ∆r→0 f  ∆r e ıθ  ∆r e ıθ = lim ∆r→0 ∆r 4 /3 cos 4 /3 θ∆r 5 /3 sin 5 /3 θ+ı∆r 5 /3 cos 5 /3 θ∆r 4 /3 sin 4 /3 θ ∆r 2 ∆r e ıθ = lim ∆r→0 cos 4 /3 θ sin 5 /3 θ + ı ... function f(z) = u + ıv =  x 3 (1+ı)−y 3 (1−ı) x 2 +y 2 for z = 0, 0 for z = 0. Show that the partial derivatives of u and v with respect to x and y exist at z =...

Ngày tải lên: 06/08/2014, 01:21

... 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 and C 3 be contours around z = 3 √ 9, ... 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) s...

Ngày tải lên: 06/08/2014, 01:21

... , converges for α > 1 and diverges for α ≤ 1. Hint, Solution 564 Example 12 .3. 2 Convergence and Uniform Convergence. Consider the series log(1 − z) = − ∞  n=1 z n n . This series converges for |z| ... expansions of 1/(1 + z). For |z| < 1, 1 1 + z = 1 +  −1 1  z +  −1 2  z 2 +  −1 3  z 3 + ··· = 1 + (−1) 1 z + (−1) 2 z 2 + (−1) 3 z 3 + ··· = 1 − z + z 2 − z...

Ngày tải lên: 06/08/2014, 01:21

... 1))  z n + d, for 2 < |z| Solution 12.29 The radius of convergence of the series for f(z) is R = lim n→∞     k 3 /3 k (k + 1) 3 /3 k+1     = 3 lim n→∞     k 3 (k + 1) 3     = 3. 622 The ... 13. 1 .3 Evaluate the integral  C cot z coth z z 3 dz where C is the unit circle about the origin in the positive direction. The integrand is cot z coth z z 3 =...

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 ... 0 y  − − y − = 0, y − (0) = y + (0), for x < 0, and define the solution, y, to be y(x) =  y + (x), for x ≥ 0, y − (x), for x ≤ 0. The initial condition for y − d...

Ngày tải lên: 06/08/2014, 01:21