... Equations
Hint 17 .11
Hint 17 .12
Hint 17 .13
Hint 17 .14
Substitute y = x
λ
into the differential equation. Consider the three cases: a
2
> b, a
2
< b and a
2
= b.
Hint 17 . 15
Hint 17 .16
Exact Equations
Hint ... conditions, y
1
(0) = 1 and y
1
(0) = 0, for the first solution become,
c
1
= 1, −ac
1
+
√
a
2
− b c
2
= 0,
c
1
= 1, c
2
=
a
√
a
2
− b
.
The c...
... ∼ 4
∞
n =1
oddn
cos(nx).
For x = nπ, this implies
0 = 4
∞
n =1
oddn
cos(nx),
13 61
-1
-0 .5 0 .5
1
-0.2
-0 .1
0 .1
0.2
1
0. 25
0 .1
0
0 .1
1
1
1
0 .5
Figure 28.8: Three Term Approximation for a Function ... =
−x − 1 for − 1 < x < 1/ 2
x for − 1/ 2 < x < 1/ 2
−x + 1 for 1/ 2 < x < 1.
13 55
1. Let S
N
be the sum of the fi...
... with
u = α (1 − v)
for special values of α and r. Find v(x) for this special case.
216 8
L
1
L
2
(L
1
x) = L
1
(L
1
l
2
− I)x
= L
1
(λx − x)
= (λ − 1) (L
1
x)
L
1
L
2
(L
2
x) = (L
2
L
1
+ I)L
2
x
= ... 2
√
1 − t
2
log (1 + t),
1
2
G
+
(t) + G
−
(t)
= ıt log (1 + t).
For t ∈ (−∞, 1) ,
G
+
(t) = ı
√
1 − t
2
+ t
(log(−t − 1) + ıπ) ,
G
−
(t) = ı
−
√...
... a and b are orthogonal, (perpendicular), or one of a and b are zero.
26
1
2
1
2
2
4
6
8
10
1
2
Figure 1. 11: Plots of f(x) = p(x)/q(x).
1. 6 Hints
Hint 1. 1
area = constant ×diameter
2
.
Hint 1. 2
A ... with vertices at (1, 1) , (4, 2), (3, 7) and (2, 3)?
Hint, Solution
Exercise 2.7
What is the volume of the tetrahedron with vertices at (1, 1, 0), (3, 2, 1) , (2, 4, 1) a...