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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)

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