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
