User nw patrick - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T02:15:16Z http://mathoverflow.net/feeds/user/15928 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/68654/how-to-solve-this-differential-equation How to solve this differential equation? NW Patrick 2011-06-23T21:02:45Z 2011-06-23T23:39:45Z <p>I want to solve this differential equation:</p> <p>$$C \cdot y(t)\frac{d}{dt} = x(t) - y(t)$$</p> <p>$x(t)$ and $y(t)$ are two ordinary functions of t, C is a constant - all in in $R$</p> <p>I am trying to solve it towards $y(t)$. The solution I am looking for looks something like this:</p> <p>$$y(t) = e^{\int{x(t)dt}} + C$$</p> <p>So $\int{x(t)dt}$ can stay - but how will the rest look like and could you show me the individual steps and name the method how to solve it? Could "variation of the parameters" be applied?</p> http://mathoverflow.net/questions/68428/why-is-the-output-of-an-lti-system-the-convolution-of-the-input-funtion-and-the-i Why is the output of an LTI system the convolution of the input funtion and the impulse response? NW Patrick 2011-06-21T19:50:01Z 2011-06-22T00:15:22Z <p>I am looking at the description of LTI systems in the time domain.</p> <p>Intuitively, I'd have guessed it would be the composition of the input function and some "system function". $$y(t) = f(x(t)) = (f\circ x)(t)$$ Where $x(t)$ is the input, $y(t)$ output and $f(x)$ a "system function".</p> <p>Why is it not that way? Could such a "system function" be found for, say, an R-C-Circuit?</p> <p>The actual output function y(t), is defined as $$y(t) = (h * x)(t)$$ Where $h(t)$ is the response to a dirac impulse. This is hard to grasp for me. Why is it so? I have looked at various explanations, drawings of rectangles becoming infinitely narrow, which I sort of understood, but it is still "hard to grasp"! I am looking for a simple explanation in one or two sentences here.</p> <p><a href="http://en.wikipedia.org/wiki/LTI_system_theory" rel="nofollow">http://en.wikipedia.org/wiki/LTI_system_theory</a></p> http://mathoverflow.net/questions/68654/how-to-solve-this-differential-equation Comment by NW Patrick NW Patrick 2011-06-24T14:13:50Z 2011-06-24T14:13:50Z $\frac{d}{dt}$ is applied to $t$, which $y(t)$ is a function of. I thought I knew what a differential equation is. @Gerald There are of course (many) other possibilities to write it down. Which notation would you prefer ? http://mathoverflow.net/questions/68654/how-to-solve-this-differential-equation Comment by NW Patrick NW Patrick 2011-06-23T21:07:44Z 2011-06-23T21:07:44Z This comes very close to what I'm trying to do but my equation is a little bit simpler: <a href="http://de.wikipedia.org/wiki/Variation_der_Konstanten" rel="nofollow">de.wikipedia.org/wiki/Variation_der_Konstanten</a> http://mathoverflow.net/questions/68428/why-is-the-output-of-an-lti-system-the-convolution-of-the-input-funtion-and-the-i/68453#68453 Comment by NW Patrick NW Patrick 2011-06-22T13:14:28Z 2011-06-22T13:14:28Z All right. I was too locked in to use $f$ as the $f$unction letter. Also, our prof has been using $\omega$, the circular frequency, which equals to $2\pi f$, mostly. -- I am aware of the advantages of doing things in the frequency domain but I'd nevertheless like to see the mess I'm avoiding, just once, to have an better understanding of what I'm doing http://mathoverflow.net/questions/68428/why-is-the-output-of-an-lti-system-the-convolution-of-the-input-funtion-and-the-i/68463#68463 Comment by NW Patrick NW Patrick 2011-06-22T01:05:20Z 2011-06-22T01:05:20Z This (together with <a href="http://www.khanacademy.org/video/introduction-to-the-convolution?playlist=Differential%20Equations" rel="nofollow">khanacademy.org/video/&hellip;</a> &amp; a variety of other documents) helped me actually understand what convolution is. -- Thanks a lot! -- // Because an R-C-Circuit can only be described by dirac- or step-function response or an ODE in the time domain, my quest for a &quot;system function&quot; f(x) must be futile, it seems. Correct? http://mathoverflow.net/questions/68428/why-is-the-output-of-an-lti-system-the-convolution-of-the-input-funtion-and-the-i/68453#68453 Comment by NW Patrick NW Patrick 2011-06-22T00:17:33Z 2011-06-22T00:17:33Z Thanks a lot. The answer is not that useful to me because I am not familiar with the term &quot;input-output operator&quot;. I find $f$ a confusing choice for the name of a constant here. Furthermore I'd prefer to look at the problem without changing to the frequency domain. As to the &quot;system function&quot;: How would it look like?