6.3.3 Example C The equation Yk+1Yk – 2yk = -2 (6.38) can be transformed into the linear equation Xk+2 2xk+1 + 2xk = 0 (6.39) by means of the substitution Yk = *k+1/xk. (6.40) The characteristic equation for equation (6.39) is p2 – 2r + 2 = 0, (6.41) and its two complex conjugate roots are r1,2 = 1+i = V2etin/4 (6.42) Therefore, the general solution of equation (6.39) is æk = 26/2[D1 cos(Tk/4)+ D2 sin(tk/4)], (6.43) and - 15D1 cos[T(k+1)/4] + D2 sin[T(k+1)/4] Dị cos(Tk/4) + D2 sin(rk/4) (6.44)
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
![6.3.3 Ехample C
The equation
Yk+1Yk – 2yk = -2
(6.38)
can be transformed into the linear equation
Xk+2 – 2xk+1+2xk = 0
(6.39)
by means of the substitution
Yk =
Xk+1/xk.
(6.40)
The characteristic equation for equation (6.39) is
p2 – 2r + 2 = 0,
(6.41)
and its two complex conjugate roots are
r1,2 = 1+i = v2e±i™/4_
(6.42)
Therefore, the general solution of equation (6.39) is
X} = 2k/2[D1 cos(tk/4)+ D2 sin(Tk/4)],
(6.43)
and
D1 cos[T(k+ 1)/4] + D2 sin[r(k +1)/4]
Dị cos(Tk/4) + D2 sin(tk/4)
Yk =
V2-
(6.44)
200
Difference Equations
Now, define the constant a such that in the interval -T/2 < a < T/2,
tan a =
D2/D1.
(6.45)
With this result, equation (6.44) becomes
V2 cos[(T/4)(k + 1) – a]
Yk =
cos(Tk/4 – a)
(6.46)
or
Yk
1- tan(ak/4 – a).
(6.47)
This is the general solution to equation (6.38).
Note that since tan(0+ 7) = tan 0, the solution has period 4; i.e., for given
constant a, equation (6.47) takes on only four values; they are
Yo = ]
tan(-a),
- tan(7/4 – a),
1- tan(7/2 – a),
1– tan(37/4 – a).
Y1 =1-
(6.48)
Y2 =
Y3
An easy calculation shows that yo = Y4.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F52e77568-6ab0-477a-b8a0-d1d660cfb901%2Fd6f9d001-565c-4eaf-b4b9-b9852ec5a681%2Faeajw5a_processed.jpeg&w=3840&q=75)
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 2 images
