# Second-Order Harmonic Motion Equation: Solve as 2 linear diff eqs

1. Oct 29, 2014

### leonmate

1. The problem statement, all variables and given/known data

Write the second-order equation for the harmonic oscillator,

y ̈ + ω2y = 0

as two linear ordinary differential equations. What is the analytic solution y(t) if ω = 2π, y(0) = 1 and y′(0) = 0?

2. Relevant equations

3. The attempt at a solution

So, I have to write some python code to solve this and a series of other questions but I'm falling at the first hundle! :(

What does it mean, as two linear ordinary differential equations, how do you do that??? I've been searching the internet for hours and I can't find what I'm looking for. I watched about an hour of khan academy second-order equations.

Could someone help me out, I can do the coding I just don't know what to put in!

2. Oct 29, 2014

### Dr.D

u = y
v = y'
so that v' = y"
Then
u' = v
v' = - w^2 u
And similarly transform the ICs.

3. Oct 29, 2014

### leonmate

ICs?

4. Oct 29, 2014

### SteamKing

Staff Emeritus
IC = initial conditions

5. Oct 29, 2014

### SteamKing

Staff Emeritus
I'm not sure what a 'hundle' is, but it sounds serious.

In any event, the OP asks you to find the analytic solution to the DE, which you should be able to do without writing any Python code.

It's not clear why you had so much difficulty with converting a second order ODE to a system of first-order equations. The standard methods of solving ODEs numerically (Euler, Runge-Kutta, etc.) are all based on solving first order equations only, thus a higher order equation must first be transformed into a system of first-order equations before a numerical solution can be obtained:

http://www.efunda.com/math/num_ode/num_ode.cfm

6. Oct 30, 2014

### Vibhor

I understand OP made a typo ,but the above line is seriously funny :p.