A general solution to a linear homogeneous differential equation with constant coefficients, obtained from its characteristic equation. The characteristic equation is a polynomial in which the ith derivative is replaced by rⁱ, which can then be solved for r to obtain all the roots of the polynomial. These roots are used to form the general solution. For example, the characteristic equation of $a_{2}y\text{'}\text{'}+a_{1}y\text{'}+a_{0}y=0$ is $a_2{r}^2+a_1{r}^1+a_0=0$. See also particular integral.