**Virginia Tech, MATH 2214**

**Excerpt:** ... on is important enough throughout the next few sections that we give it a name: De nition: If ay + by + c y = is a second-order homogeneous equation, then we call the polynomial ak + bk + c = the ** characteristic polynomial ** of the di erential equation. (Our textbook uses instead of k for the variable in the ** characteristic polynomial **.) e ** Characteristic Polynomial ** e ** characteristic polynomial ** for a second-order homogeneous equation is simply the equation of a parabola. e number of roots of a parabola must be zero, one, or two, depending on how o en the parabola crosses the x-axis: If the quadratic formula, k= -b b - ac , a is used to nd the roots, the discriminant (b - ac) identi es the number of roots: a positive discriminant means that both roots are real numbers; a negative discriminant means that both roots are complex numbers. When the discriminant is zero, both roots are real, but they're the same. Two (di erent) real roots When the ** characteristic polynomial ** has two real roots k ...