## FANDOM

17 Pages

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

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