# Taylor expansion of ODE?

1. Apr 7, 2014

### tssuser

I'm confused by problem 2.31 in mathematical tools for physics.

Problem:
2.31 The Doppler effect for sound with a moving source and for a moving observer have different formulas. The Doppler
effect for light, including relativistic effects is different still. Show that for low speeds they are all about the same.

$f' = f \frac{v - v_0}{v}$, $f' = f \frac{v}{v+v_s}$, $f' = f \sqrt{\frac{1-v/c}{1+v/c}}$

The symbols have various meanings: v is the speed of sound in the first two, with the other terms being the velocity
of the observer and the velocity of the source. In the third equation c is the speed of light and v is the velocity of the
observer. And no, 1 = 1 isn't good enough; you should get these at least to first order in the speed.

Solution:
From the selected solutions:
$f' = f(1-v_0/v)$, $f' = f(1-v_s/v)$, $f'=f(1-v/c)$

Question:
Clearly I'm supposed to do a tailor expansion of something, but I'm unsure of which part of the original differential equation I'm supposed to expand. Also, whichever part I do expand I end up with a different result than the given solution, which makes me think I'm interpreting the equation wrong. My interpretation is:
$f'(x) = \frac{v - v_0}{v} f(x)$

Thanks for any help clearing this up.

2. Apr 7, 2014

### Jilang

Yes, a Taylor expansion is the way to go. These are not differential equations though. The primes merely denote frequencies in different reference frames.

3. Apr 7, 2014

### tssuser

Thanks Jilang, that cleared things up for me. I suppose this is a difference between a "Math for Physics" text and a pure mathematics one.

I'm new to the forum, is there a standard for marking questions and posts as [solved] ?

4. Apr 7, 2014

### SammyS

Staff Emeritus
Hello tssuser. Welcome to PF !