Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Just a simple undergrad physics student asking them mathematicians.

I have a very simple 2nd order homogeneous DE of the form:


So, a solution will be of the form $$y(t)=c_1 e^{+iat}+c_2 e^{-iat} \quad (1)$$

which you can rewrite to $$y(t)=d_1\cos(at)+d_2 \sin(at) \quad (2)$$

The final answer of the problem has the form of $C \sin(at+d)$, where $d$ is a phase term. No initial conditions are given.

My question is, can eq. (2) be rewritten in some way that only a sine term emerges, with a constant $d$ appearing in there? I looked at some trig. relations and couldn't see a quick way. I think the initial conditions have to be used. What do you think?

(A teaching assistent told me it could be done, but I distrust her advice a little since she has given me wrong answers before)

share|cite|improve this question

Expand on the trig expression you get from the answer:

$$\sin(at+d) = \sin(at) \cos(d) + \cos(at) \sin(d)$$

So you can take eqn.(2) and factor out the constant $C = \sqrt{d_1^2 + d_2^2}$ and then pick $d$ so that $\cos(d) = d_1/C$ and $\sin(d) = d_2/C$.

share|cite|improve this answer
WE did it at the same time! – Babak S. Feb 27 '13 at 13:43
@BabakS. :) yes, indeed – gt6989b Feb 27 '13 at 13:44

As far as I know for a suitable $\eta$, we can have $a\sin(x)+b\cos(x)=\sqrt{a^2+b^2}\sin(x+\eta)$

share|cite|improve this answer
Yay for $\eta$ and Babak +1 – amWhy Feb 28 '13 at 0:15

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.