An Euler-Cauchy equation is
where b and c are constant numbers. Let us consider the change of
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:
- Write down the characteristic equation
- 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.
- If the roots r1 and r2 are equal (r1 = r2), then
the general solution of (EC) is
- If the roots r1 and r2 are complex numbers, then the
general solution of (EC) is
Example: Find the general solution to
Solution: First we recognize that the
equation is an Euler-Cauchy equation, with b=-1 and c=1.
- Characteristic equation is
r2 -2r + 1=0.
- Since 1 is a double root, the general solution is
[First Order D.E.]
[Second Order D.E.]
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