# Thread: Simple equation - question

1. ## Simple equation - question

Hi!

I just started up some repetition but got stuck when I looked at my techers answers.

I have this equation: x´´ + (k/m)x = 0

Then I get from the auxilliary solution that: x = Acos(sqrt(k/m))

However my teacher has written: x = Acos(sqrt(k/m)+a)

So he has some phase-constant a, and here is my question.

I see that this isnt that strange but how did it enter? Can someone show it to mathematically? Logically I think it makes sence but I'd like to see how.

2. What variable is $\displaystyle x$ a function of? $\displaystyle x$ can not possibly equal $\displaystyle A\cos{\left(\sqrt{\frac{k}{m}}\right)}$ since this is a number, not a function.

3. x is a function of time, because the DE is the equation of motion for a mass on a spring undergoing simple harmonic motion. There should be $\displaystyle t$'s multiplying the square roots inside the trig functions.

I don't know what your "auxiliary" solution is; however, without initial conditions, it is incorrect. Your teacher's answer is correct. You can either write the homogeneous solution as

$\displaystyle x(t) = A \cos(t\sqrt{k/m} + a)$, with the phase angle, or as
$\displaystyle x(t) = A \cos(t\sqrt{k/m}) + B \sin(t\sqrt{k/m}).$

It can be shown that these two approaches are equivalent.

Incidentally, I should point out that in the physics world, at least, there is a convention that primes $\displaystyle (r'')$ are usually reserved for spatial derivatives, and dots $\displaystyle (\ddot{r})$ are usually reserved for temporal derivatives.

4. Thanks for the answers Ackbeet! Yes its true I forgot to write the t inte my equation.

Is there anyone who would like to show,mathematically, that these 2 statements are the same?

, with the phase angle, or as

5. Use the addition of angles formula for the cosine. You'll absorb the cos(a) and sin(a) into the A and B constants of integration, and then you're done. Going the other way, you can simply re-define the A and B of the second equation to have the cos(a) and sin(a) that you need, use the addition of angles formula and simplify into the first equation. Incidentally, the A in the second equation is not the same A as in the first equation. You just need to have two constants (the two constants of integration you should always get for a second-order ODE).

It all comes down to the addition of angles formula for the cosine.

6. Thanks for answer again Ackbeet. Very simple when you know what to do.

7. You're very welcome. Have a good one!