2|C|k°]Re{el°• = k°{A cos(bln k)+B sin(b ln k)} 3 In k + B sin 2 [(4)-}. /3 In k 2 k1/2 { A cos
Family of Curves
A family of curves is a group of curves that are each described by a parametrization in which one or more variables are parameters. In general, the parameters have more complexity on the assembly of the curve than an ordinary linear transformation. These families appear commonly in the solution of differential equations. When a constant of integration is added, it is normally modified algebraically until it no longer replicates a plain linear transformation. The order of a differential equation depends on how many uncertain variables appear in the corresponding curve. The order of the differential equation acquired is two if two unknown variables exist in an equation belonging to this family.
XZ Plane
In order to understand XZ plane, it's helpful to understand two-dimensional and three-dimensional spaces. To plot a point on a plane, two numbers are needed, and these two numbers in the plane can be represented as an ordered pair (a,b) where a and b are real numbers and a is the horizontal coordinate and b is the vertical coordinate. This type of plane is called two-dimensional and it contains two perpendicular axes, the horizontal axis, and the vertical axis.
Euclidean Geometry
Geometry is the branch of mathematics that deals with flat surfaces like lines, angles, points, two-dimensional figures, etc. In Euclidean geometry, one studies the geometrical shapes that rely on different theorems and axioms. This (pure mathematics) geometry was introduced by the Greek mathematician Euclid, and that is why it is called Euclidean geometry. Euclid explained this in his book named 'elements'. Euclid's method in Euclidean geometry involves handling a small group of innately captivate axioms and incorporating many of these other propositions. The elements written by Euclid are the fundamentals for the study of geometry from a modern mathematical perspective. Elements comprise Euclidean theories, postulates, axioms, construction, and mathematical proofs of propositions.
Lines and Angles
In a two-dimensional plane, a line is simply a figure that joins two points. Usually, lines are used for presenting objects that are straight in shape and have minimal depth or width.
Explain the determine red
![it can be very useful to compare their asymptotic behaviors. To do so, rewrite
equation (8.83) as
T(k+r)
I(k)
Yk
2 Re
(8.86)
where r = ri =
r*, and let C =
|C|e", where |C| is the magnitude of the
complex constant C and 0 is its constant phase. A property of the gamma
function is that for large k
T(k +r)
T(k)
k".
k-large
(8.87)
Therefore,
Yk
k-large
• 2Re{\C\e®k«+ib},
(8.88)
where
3
1
ib = r = -+
2
(8.89)
2
and
Yk
k-large
+[2|C\k"]Re{e'(b\nk+0)}
k° {A cos(b ln k) +B sin(b ln k)}
[(4).
kl/2
A cos
In k + B sin
In k
(8.90)
where A and B are two real arbitrary constants. As a comparison of y(x),
equation (8.81), and yk, equation (8.90) shows that they have exactly the
same asymptotic behavior.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F06b6425a-ab8a-4368-90d9-64663800e88c%2F81ebff56-ca51-4872-a692-b2a28d97573a%2Fiiced44_processed.jpeg&w=3840&q=75)
![8.3.2
Example B
Consider the following Cauchy-Euler differential equation
d²y
+ y = 0.
dx2
(8.77)
Its characteristic equation is
r(r – 1) +1= r2 – r +1= 0,
(8.78)
-
with solutions
1
= ra
i,
= V-1.
(8.79)
ri =
+
2
To obtain the general solution, the quantity x"1 must be calculated. This is
done as follows
(a) exp
V3
x"1
In x
(8.80)
= x
= X
• x
Therefore, y(x) is given by the expression
y(x) = x
A cos
In x + B sin
In x
(8.81)
Note that for x > 0, the solution oscillates with increasing amplitude.
The corresponding discrete version of equation (8.77) is
k(k + 1)A²yk + Yk
0,
(8.82)
and its characteristic equation is that given in equation (8.78). Therefore, the
general solution is
,r(k+r)
I(k)
T(k +r*)
+ C*
T(k)
Yk =
(8.83)
where r = r1 and C is an arbitrary complex number. Observe that the manner
in which the right side of equation (8.83) is written insures real values for yk.
This depends also on the fact that the gamma function T(z) is real-valued,
i.e., for z = x+ iy, I'(z*) = [T'(2)]*. The integral representation of the gamma
function allows this to be easily demonstrated, i.e.,
I(2) = | e-t-1dt.
(8.84)
Jo
Finally, while it is not to be expected that y(x), equation (8.81), and yk,
equation (8.83) have "exactly" the same mathematical structure for all x and
k, where the correlation between these variables is
x → Xk =
(Ax)k, Ax = 1,
(8.85)
%3D](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F06b6425a-ab8a-4368-90d9-64663800e88c%2F81ebff56-ca51-4872-a692-b2a28d97573a%2Fx6u2muq_processed.jpeg&w=3840&q=75)
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