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

I've been studying electronics, where they make great use of the relationship between the sine and exponential functions ($e^{i \omega t} = \cos{\omega t} + i \sin \omega t)$. This relationship is confusing to me, so I started digging into it, and thinking about how they have similar definitions, in terms of differential equations.

$f(x) = e^x$ is the solution to this differential equation:

$$ f'(x) = f(x) $$

and $f(x) = \sin x$ is a solution to this similar equation:

$$ f'(x) = f(x + \pi/2) $$

I wanted to see solutions to the following, for other values of constant $k$.

$$ f'(x) = f(x + k) $$

but my differential equation solving skills are non-existent. So my main question is: What are these functions, and what do their graphs look like? A secondary question is: Do you know how to write the bit of code necessary to solve that third DE (for some value of $k$) using sage or wolfram alpha? I have sage but don't know what to write.

share|cite|improve this question
See my answer here. – Mhenni Benghorbal Nov 23 '12 at 20:44
There is a typo in the last equation, $k\mapsto x$. – NikolajK Nov 23 '12 at 21:09
Thanks, fixed it. – Rob N Nov 23 '12 at 21:50
Another interesting feature, following from you differential equation, that I realized when thinking about your problem is the implication for $f^{(n)}(x)$, namely shift to $f(x+nk)$. – NikolajK Nov 23 '12 at 23:12
up vote 6 down vote accepted

They are all solutions of some $$ f'' + A f' + B f = 0 $$ with constants $A,B.$ The constants can be real numbers for $e^x, \sin x,$ but the full story allows them complex as needed.

Don't get sidetracked by the first order delay differential equations $f'(x) = f(x+k).$ The reason that appears is that there are identities for $\sin (x+k)$ and $\cos(x+k).$

As a simple example, $$ f'' + 2 f' + 2 f = 0 $$ has solutions $$ e^{-x} \sin x , \; \; e^{-x} \cos x $$ as (real-valued) solutions. So your $f(x)$ could be $$ f(x) = C e^{-x} \cos x + D e^{-x} \sin x $$ with real constants $C,D.$ Note that damping effect of the $e^{-x},$ which says that any such $f$ oscillates but goes fairly rapidly to $0.$ This is the type of phenomenon worked with in an automobile spring/shock absorber system.

The similar but unusual $$ f'' + 2 f' + f = 0 $$ has solutions $$ e^{-x} , \; \; x e^{-x} $$ as (real-valued) solutions. So your $f(x)$ could be $$ f(x) = C e^{-x} + D x e^{-x} $$ with real constants $C,D.$

share|cite|improve this answer
If I could I would accept more than one answer; all were helpful. The others address the delay differential equation; but as you said, that would be a sidetrack given my immediate goal of understanding the electric circuits. Thanks everyone! – Rob N Nov 30 '12 at 0:08

There's a good reason why you're finding yourself unable to find solutions in the latter cases. Those are examples of delay differential equations and they have a theory that's substantially more complicated than that of ODEs.

share|cite|improve this answer

A year ago I asked a similar question here. So let me generalize GEdgars nice answer. Choose a $k$ and consider

$$f(x):=\sum_n a_n\ \text{e}^{M_n x/k},$$

with some magic numbers $M_n$ fulfilling the relation $\text{e}^{M_n}=M_n/k$. These are related to the Lambert W function as explained in the question I linked you to above. Then

$$f(x+k)=\sum_n a_n\ \text{e}^{M_n x/k}\ \text{e}^{M_n}=\sum_n a_n\ \text{e}^{M_n x/k}\ M_n/k=f'(x).$$

share|cite|improve this answer

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.