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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.


See also

Euler's Equations of Inviscid Motion

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References

Valiron, G. The Geometric Theory of Ordinary Differential Equations and Algebraic Functions. Brookline, MA: Math. Sci. Press, 1950.Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 120, 1997.

Referenced on Wolfram|Alpha

Euler Differential Equation

Cite this as:

Weisstein, Eric W. "Euler Differential Equation." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/EulerDifferentialEquation.html

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