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Euler Differential Equation

The general nonhomogeneous differential equation is given by

 (1)

and the homogeneous equation is

 (2)
 (3)

Now attempt to convert the equation from

 (4)

to one with constant coefficients

 (5)

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

 (6)
 (7)

Let and define

 (8) (9) (10) (11)

Then is given by

 (12) (13) (14)

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

 (15)

Define

 (16) (17) (18) (19)

and

 (20) (21)

The solutions are

 (22)

In terms of the original variable ,

 (23)

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

 (24)

(Valiron 1950, p. 201) and

 (25)

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

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.

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Euler Differential Equation

Cite this as:

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