Advanced Mathematical Methods for Scientists and Engineers Episode 3 Part 5 pdf
Advanced Mathematical Methods for Scientists and Engineers Episode 3 Part 5 pdf
... 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
9 45
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
/3...
Advanced Mathematical Methods for Scientists and Engineers Episode 1 Part 5 pdf
... +
√
87
2 /3
− 6
−1 /3
9 +
√
87
1 /3
2
≈ (0 .58 9 755 , 0, 0 .34 781).
The closest point is shown graphically in Figure 5. 10.
-1
-0 .5
0
0 .5
1
-1
-0 .5
0
0 .5
1
0
0 .5
1
1 .5
2
-1
-0 .5
0
0 .5
1
0
0 .5
1
1 .5
2
Figure ... =
6
−2 /3
9 +
√
87
2 /3
− 6
−1 /3
9 +
√
87
1 /3
≈ 0 .58 9 755 .
Thus the closest point to (1, 0, 0) on the paraboloid is
6
−2 /3...
Advanced Mathematical Methods for Scientists and Engineers Episode 4 Part 5 pdf
... sin(nπx) dx
=
4
(nπ)
2
sin(nπ/2)
=
4
(nπ)
2
(−1)
(n−1)/2
for odd n
0 for even n.
1 35 6
-3
-2
-1 1
2
3
-1 .5
-1
-0 .5
0 .5
1
1 .5
Figure 28 .5: Fourier Sine Series.
The functions {. . . ,
e
−ıx
, 1,
e
ıx
,
e
ı2x
, ... -1
-0 .5 0 .5
1
-1
-0 .5
0 .5
1
-1
-0 .5 0 .5
1
-0.4
-0.2
0.2
0.4
Figure 28.7: Three Term Approximation for a Function with Jump Discontinuities and a C...
Advanced Mathematical Methods for Scientists and Engineers Episode 6 Part 5 pdf
... are
λ
1
≈ 0.678298
λ
2
≈ 7.27 931
λ
3
≈ 24. 930 2
λ
4
≈ 54 . 25 93
λ
5
≈ 95 .30 57
The eigenfunctions are,
φ
n
(x) = v
(1; λ
n
)u(x; λ
n
) − u
(1; λ
n
)v(x; λ
n
).
Solution 48 .32
1. First note that
sin(kx) ... equation is
φ(x) =
a + bx for λ = 0
a cos
√
λx
+ b sin
√
λx
for λ > 0
a cosh
√
−λx
+ b sinh
√
−λx
for λ < 0
We see that for...
Advanced Mathematical Methods for Scientists and Engineers Episode 1 Part 1 pdf
... . . . . 150 7
32 The Fourier Transform 1 53 9
32 .1 Derivation from a Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 53 9
32 .2 The Fourier Transform . . ... q(x).
Hint, Solution
12
33 The Gamma Function 16 05
33 .1 Euler’s Formul a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 05
33 .2 Hank...
Advanced Mathematical Methods for Scientists and Engineers Episode 2 Part 5 pps
... 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
3
+ ···
55 8
For |z| > 1,
1
1 ... is uniformly convergent for |z| ≤ r < 1.
54 5
12.2.2 Uniform Convergence and Continuous Functions.
Consider a series f(z) =
∞
n=1
a
n
(z) that is uniformly convergent...
Advanced Mathematical Methods for Scientists and Engineers Episode 2 Part 7 pdf
... 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 ... cosh z
z
3
sin z sinh z
=
1 −
z
2
2
+
z
4
24
− ···
1 +
z
2
2
+
z
4
24
+ ···
z
3
z −
z
3
6
+
z
5
120
− ···
z +
z
3
6
+
z
5
120
+ ···
=
1 −
z
4...
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 ... < 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
−
demands that the solution be continuous.
Solvin...
Advanced Mathematical Methods for Scientists and Engineers Episode 3 Part 2 potx
... =
1
2
1
1
e
−t
+
1
2
1
5
e
3t
For large t, the solution looks like
x ≈
1
2
1
5
e
3t
.
Both coordinates tend to infinity.
Figure 15. 1 shows some homogeneous solutions in the phase plane.
Example 15. 2.2 (mathematica/ode/systems/systems.nb) ... 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 mak...
Advanced Mathematical Methods for Scientists and Engineers Episode 3 Part 3 pptx
... =
2 0 1
0 2 0
0 1 3
2. Solve
dx
dt
= Ax + g(t), x(0) = 0
868
15. 6 Hints
Hint 15. 1
Hint 15. 2
Hint 15 .3
Hint 15. 4
Hint 15. 5
Hint 15. 6
Hint 15. 7
Hint 15. 8
Hint 15. 9
Hint 15. 10
870
where S is ... =
5 − λ 3 −2
8 5 − λ −4
−4 3 3 −λ
= −λ
3
+ 3
2
− 3 + 1 = −(λ − 1)
3
λ = 1 is an eigenvalue of multiplicity 3. The rank of the null space of A...
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