An Euler-Cauchy equation is

where

Then we have

The 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
*r*_{1}and*r*_{2}are distinct real numbers, then the general solution of (*EC*) is given by

*y*(*x*) =*c*_{1}|*x*|^{r1}+*c*_{2}|*x*|^{r2}.

- (2)
- If the roots
*r*_{1}and*r*_{2}are equal (*r*_{1}=*r*_{2}), then the general solution of (*EC*) is

- (3)
- If the roots
*r*_{1}and*r*_{2}are complex numbers, then the general solution of (*EC*) is

where and .

**Example:** Find the general solution to

- 1
- Characteristic equation is
*r*^{2}-2*r*+ 1=0. - 2
- Since 1 is a double root, the general solution is

**
**

Do you need more help? Please post your question on our S.O.S. Mathematics CyberBoard.

Contact us

Math Medics, LLC. - P.O. Box 12395 - El Paso TX 79913 - USA

users online during the last hour