# Modes and the Characteristic Equation

Introduction of the Characteristic Equation

Several presumptions are made in order to solve

{$$ my^{\prime\prime} + by^{\prime} + ky = 0 $$}.

This form emphasizes the application to spring-damper problems in which *m* is mass, *k* is the spring coefficient from Hooke's law, and *b* is a coefficient of damping. These are taken to be constant over the applicable conditions. Usually tacit is that the independent variable is *t* which is time in most physical problems. Dividing through by *m* results in the standard form:

{$$ y^{\prime\prime} + Ay^{\prime} + By = 0 $$}.

Some results depend on A and B being real numbers, which in physical modeling they should be.

From this we find the characteristic equation by substituting {$ e^{rt} $} for *y*.

{$$ \begin{align} y^{\prime\prime} + Ay^{\prime} + By &= 0 \cr ( e^{rt})^{\prime\prime} + A( e^{rt})^{\prime} + B( ) &= 0 \cr r^2e^{rt} + Are^{rt} + Be^{rt} &= 0 \cr \left(r^2 + Ar + B\right)e^{rt} &= 0 \cr r^2 + Ar + B &= 0 \cr \end{align}$$}.

This is a quadratic equation, because the differential equation is second order. It has two roots, and the possibilities are:

Roots | Case | Roots | Solution |
---|---|---|---|

Two real and unequal roots | overdamped | {$ r_1,r_2 $} | {$ y = c_1e^{r_1 t} + c_2e^{r_2 t} $} |

Two real and equal roots | underdamped | {$ r_1 = r_2 $} | {$ y = c_1e^{r_1 t} + c_2te^{r_1 t} $} |

Two complex roots | critically damped | {$ r=\alpha \pm \beta i $} | {$ y = e^{\alpha t}[ c_1\cos(\beta t) + c_2\sin(\beta t)] $} |

<< Unit II: Second Order Constant Coefficient Linear Equations | Trail MIT 18.03SC Differential Equations | Damped Harmonic Oscillators >>

*Sources:*

- Lec 9 | MIT 18.03 Differential Equations, Spring 2006
- First-order Constant Coefficient Linear ODE's | MIT 18.03SC Differential Equations, Fall 2011
- Homogeneous Constant Coefficient Equations: Real Roots | MIT 18.03SC Differential Equations
- Problem Solving Videos

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