# Homogeneous second-order linear ordinary differential equation with constant coefficients

## Equation

Governing equation:

 (1)

where and are real constants.

## Solution

Let and be the roots of the characteristic equation

 (2)

Then

 (i) If real, then (3) (ii) If , then (iii) If , (), then

References: Bubenik 1997.

## Derivation

The solution is expected in the form of and the goal is to determine .

Express derivatives of y:

Substituting back to the original equation yields:

Since , then the following characteristic equation must be satisfied:

The solution of the quadratic characteristic equation is:

### Two Real Roots

If , then the characteristic equation has two real roots and the solution is:

### One Real Root

If , then the characteristic equation has a single real root and the solution is:

### No Real Root

If , then the characteristic equation has a two complex roots.

To Do: derivation of solution for second order linear ordinary differential equation - clean up stuff. (who: user:ok; priority: 100; hours: 0) (all) (cat)

## Usage in Structural Engineering

##### Actions    