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Matematik Seçkileri
=> Ajima-Malfatti Points
=> Kissing Number
=> Quaternion
=> Lotka-Volterra Equations
=> Euler Differential Equation
=> Dilogarithm
=> Abelian Category
=> Base
=> Steenrod Algebra
=> Gamma Function
=> Bessel Functions
=> Jacobi Symbol
=> Quadratic Curve Discriminant
=> Illumination Problem
=> Sylvester's Four-Point Problem
=> Triangle Interior
=> 6-Sphere Coordinates
=> Mordell Curve
=> Zermelo-Fraenkel Axioms
=> Peano's Axioms
=> De Morgan's Laws
=> Kolmogorov's Axioms
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Euler Differential Equation

The general nonhomogeneous differential equation is given by

 x^2(d^2y)/(dx^2)+alphax(dy)/(dx)+betay=S(x),
(1)

and the homogeneous equation is

 x^2y^('')+alphaxy^'+betay=0
(2)
 y^('')+alpha/xy^'+beta/(x^2)y=0.
(3)

Now attempt to convert the equation from

 y^('')+p(x)y^'+q(x)y=0
(4)

to one with constant coefficients

 (d^2y)/(dz^2)+A(dy)/(dz)+By=0
(5)

by using the standard transformation for linear second-order ordinary differential equations. Comparing (3) and (5), the functions p(x) and q(x) are

 p(x)=alpha/x=alphax^(-1)
(6)
 q(x)=beta/(x^2)=betax^(-2).
(7)

Let B=beta and define

z = B^(-1/2)intsqrt(q(x))dx
(8)
= beta^(-1/2)intsqrt(betax^(-2))dx
(9)
= intx^(-1)dx
(10)
= lnx.
(11)

Then A is given by

A = (q^'(x)+2p(x)q(x))/(2[q(x)]^(3/2))B^(1/2)
(12)
= (-2betax^(-3)+2(alphax^(-1))(betax^(-2)))/(2(betax^(-2))^(3/2))beta^(1/2)
(13)
= alpha-1,
(14)

which is a constant. Therefore, the equation becomes a second-order ordinary differential equation with constant coefficients

 (d^2y)/(dz^2)+(alpha-1)(dy)/(dz)+betay=0.
(15)

Define

r_1 = 1/2(-A+sqrt(A^2-4B))
(16)
= 1/2[1-alpha+sqrt((alpha-1)^2-4beta)]
(17)
r_2 = 1/2(-A-sqrt(A^2-4B))
(18)
= 1/2[1-alpha-sqrt((alpha-1)^2-4beta)]
(19)

and

a = 1/2(1-alpha)
(20)
b = 1/2sqrt(4beta-(alpha-1)^2).
(21)

The solutions are

 y={c_1e^(r_1z)+c_2e^(r_2z)   (alpha-1)^2>4beta; (c_1+c_2z)e^(az)   (alpha-1)^2=4beta; e^(az)[c_1cos(bz)+c_2sin(bz)]   (alpha-1)^2<4beta.
(22)

In terms of the original variable x,

 y={c_1|x|^(r_1)+c_2|x|^(r_2)   (alpha-1)^2>4beta; (c_1+c_2ln|x|)|x|^a   (alpha-1)^2=4beta; |x|^a[c_1cos(bln|x|)+c_2sin(bln|x|)]   (alpha-1)^2<4beta.
(23)

Zwillinger (1997, p. 120) gives two other types of equations known as Euler differential equations,

 y^'=+/-sqrt((ay^4+by^3+cy^2+dy+e)/(ax^4+bx^3+cx^2+dx+e))
(24)

(Valiron 1950, p. 201) and

 y^'+y^2=alphax^m
(25)

(Valiron 1950, p. 212), the latter of which can be solved in terms of Bessel functions.

 

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