# Euler-Cauchy Equations

An Euler-Cauchy equation is

where b and c are constant numbers. Let us consider the change of variable

x = et.

Then we have

The equation (EC) reduces to the new equation

We recognize a second order differential equation with constant coefficients. Therefore, we use the previous sections to solve it. We summarize below all the cases:
(1)
Write down the characteristic equation

(2)
If the roots r1 and r2 are distinct real numbers, then the general solution of (EC) is given by

y(x) = c1 |x|r1 + c2 |x|r2.

(2)
If the roots r1 and r2 are equal (r1 = r2), then the general solution of (EC) is

(3)
If the roots r1 and r2 are complex numbers, then the general solution of (EC) is

where and .

Example: Find the general solution to

Solution: First we recognize that the equation is an Euler-Cauchy equation, with b=-1 and c=1.
1
Characteristic equation is r2 -2r + 1=0.
2
Since 1 is a double root, the general solution is

[Differential Equations] [First Order D.E.] [Second Order D.E.]
[Geometry] [Algebra] [Trigonometry ]
[Calculus] [Complex Variables] [Matrix Algebra]

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Author: Mohamed Amine Khamsi