# What exactly is a 2nd order differential equation?

1. Oct 5, 2012

### Venomily

A first order DE models the rate of change, e.g. when decay is proportional to time we have the DE: dM/dt = -K.M; this is describing that rate of change mathematically. Am I correct in saying that a 2nd order DE describes the rate of rate of change?

Also, can anyone explain any application of 2nd order DEs to me? I understand it mathematically, but I am interested in how it works in practice like that decay example I pointed out above, hence this post.

2. Oct 5, 2012

### D H

Staff Emeritus
Classical physics is chock full of second order ODEs. F=ma, for example.

3. Oct 5, 2012

### Staff: Mentor

Not to mention electrical circuits with an inductor, resistor, and capacitor.

4. Oct 5, 2012

### Venomily

Thanks, but can you go through an example with me? actually point out a real life application (which you guys did) but also deriving a 2nd order DE to model it step by step.

5. Oct 5, 2012

### Staff: Mentor

6. Oct 10, 2012

### HallsofIvy

Staff Emeritus
If you throw a ball directly upward with initial speed 10 m/s, the basic physic law is "force= mass times acceleration". In this case, the only force (neglecting air resistance) is gravity: -mg. Since acceleration is the second derivative of the position function, taking x to be the height above the ground at time t, we have the differential equation
$$m\frac{d^2x}{dt^2}= -mg$$
with initial conditions x(0)= 0, x'(0)= 10.