Advanced Mathematical Methods for Scientists and Engineers Episode 3 Part 8 ppsx
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...
Advanced Mathematical Methods for Scientists and Engineers Episode 3 Part 6 ppsx
... 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...
Advanced Mathematical Methods for Scientists and Engineers Episode 1 Part 8 ppt
... 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...
Advanced Mathematical Methods for Scientists and Engineers Episode 2 Part 4 ppsx
... 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 =...
Advanced Mathematical Methods for Scientists and Engineers Episode 2 Part 8 pot
... 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...
Advanced Mathematical Methods for Scientists and Engineers Episode 3 Part 1 potx
... 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...
Advanced Mathematical Methods for Scientists and Engineers Episode 3 Part 2 potx
... 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...
Advanced Mathematical Methods for Scientists and Engineers Episode 3 Part 3 pptx
... 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
=
...
Advanced Mathematical Methods for Scientists and Engineers Episode 3 Part 4 pot
... 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...
Advanced Mathematical Methods for Scientists and Engineers Episode 3 Part 5 pdf
... 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
/...
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