# Ordinary Differential Equations/Homogeneous second order equations

## Introduction[edit | edit source]

The general form of order equation is

**linear non-homogeneous**if the equation can be written in the form

**linear homogeneous**if, in addition to being linear non-homogeneous,

## Method 1: Characteristic equation[edit | edit source]

If the equation is linear homogeneous and further are constant, then the equation is referred to as a **constant-coefficients** equation:

### Method formal steps[edit | edit source]

- We assume that the solution is of the form (this is called making an ansatz). This gives which equation is called the
**characteristic equation**. - So to solve the above ODE, it suffices to find the two roots .
- Then the general solution is of the form:

### Example-presenting the method[edit | edit source]

Consider a mass hanging at rest on the end of a vertical spring of length , spring constant and damping constant . Let denote the displacement, in units of feet, from the equilibrium position. Note that since represents the amount of displacement from the spring's equilibrium position (the position obtained when the downward force of gravity is matched by the will of the spring to not allow the mass to stretch the spring further) then should increase downward. Then by Newton's Third Law one can obtain the equation

where is any external force, which for simplicity we will assume to be zero.

- First we obtain the characteristic equation:
- Suppose that and then we obtain the roots , .
- Therefore, the general solution will be
- Further if we obtain :

### Examples[edit | edit source]

- Consider the IVP

- We obtain the characteristic equation and so the general solution will be
- Using the initial conditions we obtain:
- Solving these two equations gives: and so the solution for our IVP is:
- Therefore, as we obtain .

- Consider the IVP

- The characteristic equation is and so the general solution will be:
- Using the initial conditions we obtain:
- Solving these two equations gives: and so the solution for our IVP is:
- Therefore, as we obtain .