Wikipedia:Reference desk/Archives/Mathematics/2010 July 4

= July 4 =

What no questions today? Is everyone taking a holiday? —Preceding unsigned comment added by 122.107.192.187 (talk) 13:02, 4 July 2010 (UTC)
 * I've done an IP lookup on the last 4 anons who posted questions here. Two are from Australia, one from UK and one from Germany. This suggests that a lot less people ascribe special significance to the 4th of July than you'd think. Also, the day is still young, of course. -- Meni Rosenfeld (talk) 14:23, 4 July 2010 (UTC)

Second Order Differential Equations
Hi. I'm currently working through a course on DEs and now have some SODEs to solve, two examples being:

$$y''+y=H(t-\pi)-H(t-2\pi)$$

subject to y(0)=y'(0)=0 and y(t), y'(t) continuous at $$\pi, 2\pi$$

and

$$y''-4y=\delta(x-a)$$

subject to y being bounded as $$|x| \to \infty$$, with $$\delta(x)$$ being the Dirac delta function.

My problems are as follows. For each SODE I can find the complementary function and with the first one, I can find the particular integral in the cases $$t<\pi$$, $$\pi2\pi$$ but I don't know how to ensure that the solution is continuous at pi and 2pi. Then, for the second one, I haven't got a clue what to try for the particular integral. Am I supposed to do it in the same fashion as I did for the Heaviside step function and consider what happens for x=a and x≠a separately? Thanks 92.11.130.6 (talk) 15:09, 4 July 2010 (UTC)

Partial Derivatives
A quick one on PDs. If we set $$x=e^{-s}\sin t$$ and $$y=e^{-s}\cos t$$ subject to u(x,y)=v(s,t), I have to find  $$\frac{\partial v}{\partial s}$$ and  $$\frac{\partial v}{\partial t}$$ in terms of x, y,  $$\frac{\partial u}{\partial x}$$ and $$\frac{\partial u}{\partial y}$$.

Via the chain rule I get that $$\frac{\partial v}{\partial s}$$ = $$-x\frac{\partial u}{\partial x} - y\frac{\partial u}{\partial y}$$ and $$\frac{\partial v}{\partial t}$$ = $$y\frac{\partial v}{\partial x} - x\frac{\partial v}{\partial y}$$. Is this correct? I then have to find $$\frac{\partial^2 v}{\partial t^2}$$. Do I do this by finding $$\frac{\partial }{\partial t}(\frac{\partial v}{\partial t}) = \frac{\partial }{\partial t}(y\frac{\partial v}{\partial x} - x\frac{\partial v}{\partial y})$$? It's quite messy when I try it and so I doubt I'm going about this the right way. Thanks 92.11.130.6 (talk) 19:21, 4 July 2010 (UTC)
 * Hint: y+ix=e&minus;s+it. Bo Jacoby (talk) 07:04, 5 July 2010 (UTC).
 * Thanks, I hadn't actually spotted that and it's quite clever, but the final part of the question is about finding a partial differential equation for v, so I'm not allowed to know what the function is yet. 92.11.130.6 (talk) 10:04, 5 July 2010 (UTC)

Never mind, think I've got this one now. 92.11.130.6 (talk) 14:37, 5 July 2010 (UTC)