5.3.6 Example F The Lagrange method applied to the equation z(k +2,€) = 42(k, l+ 1) (5.113) gives 1? 4μ or λι 211/2 and A2 -2µ/2. Therefore, special solutions are 21(k, l) = (2µ'/2)kue (5.114) and 22(k, l) = (–2µ'/2)* e°. (5.115) Multiplying these two equations by C1(u) and C2(u) and summing over u gives the general solution z(k, e) = 2*[f(k+ 20) + (–1)*g(k+20)], (5.116) where f and g are arbitrary functions of k + 2l. Likewise, the separation-of-variables method, zp(k, l) equations CkDe, gives the Ck+2 4Ck De+1 De (5.117) where we have written the "separation constant" in the form a2. The solutions to equations (5.117) allow us to determine zp(k, l); it is žp(k, l) = [A1a*+2¢+ A2(-)*a*+2¢j2k, (5.118) where A1 and A2 are arbitrary constants. Summing over a gives the general solution expressed by equation (5.116).
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
![5.3.6 Еxample F
The Lagrange method applied to the equation
z(k + 2, e) = 42(k, l+ 1)
(5.113)
gives 12
4µ or A1
211/2 and A2
-2µ'/2. Therefore, special solutions
are
21 (k, l) = (2µ/2)*°
(5.114)
and
22(k, l) = (-2µ/2)*µ".
(5.115)
Multiplying these two equations by C1(u) and C2(µ) and summing over u
gives the general solution
z(k, l) = 2*[f(k + 2l) + (–1)*g(k + 20)],
(5.116)
where f and g are arbitrary functions of k+ 2l.
Likewise, the separation-of-variables method, zp(k, l) = C;De, gives the
equations
De+1
Ck+2
4Ck
(5.117)
De
where we have written the "separation constant" in the form a?. The solutions
to equations (5.117) allow us to determine z,(k, l); it is
k+2l
žp(k, l) = [A1a*+24 + A2(-)*a*+2¢]2*,
(5.118)
where A1 and A2 are arbitrary constants. Summing over a gives the general
solution expressed by equation (5.116).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9ff41fc3-4717-401c-a6f6-d444dc924011%2F204526f1-d721-4bd5-bfbc-39141b20ebc5%2Fwakr8i_processed.jpeg&w=3840&q=75)
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