Advanced Mathematical Methods for Scientists and Engineers Episode 3 Part 8 ppsx

... 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 particul...

Ngày tải lên: 06/08/2014, 01:21

... 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 the equation y  = y y 2 + 2x . Interchanging the dependent and independent ... becomes u  c 2 u  c + u  c 3 ξ u + u  c 3 ξ 2 = 0 u  u  + u  ξ u + u  ξ 2 = 0. We see that this equation is also equidimensional-in-x. 995 18. 9 Hints...

Ngày tải lên: 06/08/2014, 01:21

... sin 3 z = 1. sin 3 z = 1 sin z = 1 1 /3 e ız − e −ız ı2 = 1 1 /3 e ız −ı2(1) 1 /3 − e −ız = 0 e ı2z −ı2(1) 1 /3 e ız −1 = 0 e ız = ı2(1) 1 /3 ±  −4(1) 2 /3 + 4 2 e ız = ı(1) 1 /3 ±  1 − (1) 2 /3 z ... See Figure 7. 18 and Figure 7.19 for plots of the real and imaginary parts of the cosine and sine, respectively. Figure 7.20 shows the modulus of the cosine and the sine...

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, ... lower and 530 3.  C √ z dz 1. Since sin (cos (z 5 )) is an analytic function inside the unit circle,  C sin  cos  z 5  dz = 0 2. 1 (z 3) (3z−1) has singulari ties at z =...

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 ... and expand in partial fractions. Hint 13. 6 Hint 13. 7 Cauchy Principal Value for Real Integrals Hint 13. 8 680 f(a) = 2π √ 1 − a 2 Complex-Valued a. We note that the...

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 1 28 1 2 3 4 32 64 96 1 28 Figure 14.2: The discret...

Ngày tải lên: 06/08/2014, 01:21

... 0, 0 for x > 0. This behavior is shown in Figure 14.10. The first graph plots the solutions for α = 1/1 28, 1/64, . . . , 1. The second graph plots the solutions for α = 1, 2, . . . , 1 28. 83 0 Solution ... 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 to examine th...

Ngày tải lên: 06/08/2014, 01:21

... 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 88 8 We see that we ... + 2I)xi = 0.   3 1 1 2 3 −1 8 −5 −1     ξ 1 ξ 2 ξ 3   =   0 0 0   If we take ξ 3 = 1 then the first two rows give us the system,  3 1 2 3  ξ 1 ξ 2  = ...

Ngày tải lên: 06/08/2014, 01:21

... that e Jt =   e 2t t e 2t 0 0 e 2t 0 0 0 e 3t   89 4 Particular Solutions. Any function, y p , that satisfies the inhomogeneous equation, L[y p ] = f (x), is called a particular solution or particular integral of ... uncoupled equation for x 1 . x  1 = 1 t x 1 x 1 = c 1 t 89 8 Solving this equation we get a result known as Abel’s formula. W (x) = c exp  −  p n−1 (x) dx  Thus regard...

Ngày tải lên: 06/08/2014, 01:21

... to 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