A homogenous equation with constant coefficients can be written in the form

$ {a_ny^{(n)}+a_{n-1}y^{(n-1)}+...+a_0y = 0} $

and can be solved by taking the characteristic equation

$ {a_nr^n+a_{n-1}r^{n-1}+...+a_1r + a_0 = 0} $

and solving for the roots, *r.*

## Distinct Real Roots

If the roots of the characteristic equation $ {r_1, r_2,...,r_n} $, are distinct and real, then the general solution to the differential equation is

$ {y(x) = c_1e^{r_1x}+c_2e^{r_2x}+...+c_ne^{r_nx}} $

## Repeated Real Roots

If the characteristic equation has repeated roots $ {r_1, r_2,...,r_k} $, then the general solution to the differential equation has the form

$ {(c_1+c_2x+c_3x^2+...+c_kx^{k-1})e^{rx}} $

## Complex Roots

If the characteristic equation has a pair of complex conjugate roots $ {a \pm bi} $, then the general solution to the differential equation has the form

$ {e^{ax}(c_1 cos(bx) + c_2 sin(bx))} $

## External References

http://en.wikipedia.org/wiki/Characteristic_equation_(calculus)