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

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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 $|x|\to \mathrm{\infty }$, 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 $t<\pi$, $\pi <t<2\pi$ and $t>2\pi$ 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^−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)