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| visits | member for | 1 year, 6 months |
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1d |
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How to solve the two dimensional Laplace's equation for certain cases? On a side note, if you loose the spacebar between the "@" and the username, the user is notified that a comment has been made for him/her. |
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1d |
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How to solve the two dimensional Laplace's equation for certain cases? There is no inconsistency whatsoever. You'll have two solutions, one inside the shell and the other outside. To solve the first one, you'll have to make all coefficients on the singularities zero ($B_0$, $C_n$, and $D_n$), and to solve the second you do the same with the other singularities ($A_n$, $B_n$, not including $A_0$). Then you glue the problem using the bounday and decay condition and that's it. As a consequence, the electric field will be discontinuous in the shell but, hey, we knew that already! |
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2d |
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How to solve the two dimensional Laplace's equation for certain cases? Why do you say it might be convenient to take the origin outside the cable? Do you have a specific example? |
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May 21 |
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Ordinary differential equations with double resonance In ODE's, resonance occurs when you force an oscillator with a periodic force where its frequency is the same as the natural frequency of the oscillator. For example $$\ddot{y} + \omega^2 y = \sin(\omega t).$$ In this case, the natural frequency of oscillation is $\omega$. I'm not sure on double resonance though. |
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May 21 |
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Chain rule Differentiation help Where are you stuck? |
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May 16 |
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Solving ODE using frobenius method. 3 coefficients There was a mistake in the recurrence relation. See the edit for details. I'm not sure what you mean by making $a_{-1} = 0$ in the beginning. The point is that there is no $a_{-1}$, that's why you have to let go the first terms of the sums before shifting the indices, which in turn closes the recurrence relation. |
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May 7 |
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Thermodynamics for math majors @Brady Classical thermodynamics are charged with mathematical formalism, way before statistical mechanics are introduced. You should take a look to Callen's Thermodynamics and an Introduction to Thermostatistics. If you go straight to statistical mechanics, you'll be missing a lot on the subject. I don't know what kind of courses are taught around the world, but to assume that physicists are math-deprived is, for the most part, a nonsense. |
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May 7 |
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Double integral question reverse order? I've edited your question. Please make sure the it's correct. Also, you can learn the basics on how to type math in this site here meta.math.stackexchange.com/questions/107/… |
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May 7 |
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integral convergence - does this integration converge Those limits cannot be right. |
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May 7 |
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Solving series solution near a regular singular point: $2xy''+y'+xy=0$ Note that $$a_n = \frac{a_{n-2}}{(n+r)(n+r-1)+(n+r)} = \frac{a_{n-2}}{(n+r)^2}.$$ This will make the deduction of the recurrence relation much easier. Also, you have to use the indicial equation to determine the value of $r$. |
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May 7 |
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Small Question about a heat conduction problem Remember that Fourier series of $f(x)$ in $x\in(0,L)$ is defined as $$f(x) = \frac{a_0}{2} + \sum_{n=1}^\infty a_n \cos \left(\frac{n\pi x}{L}\right) + b_n \sin\left(\frac{n\pi x}{L}\right)$$ where $$a_n(x) = \frac{2}{L}\int_0^L f(x) \cos\left(\frac{n\pi x}{L}\right)dx$$ and $$b_n = \frac{2}{L}\int_0^L f(x) \sin\left(\frac{n\pi x}{L}\right)dx.$$ |
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May 7 |
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Second-order nonlinear ODE with Dirac Delta No problem. If you are really interested on what's behind delta functions in ode's, I recommend you to read the chapter on Green's function from Friedman's Principles and Techniques of Applied Mathematics. It is a fantastic introduction to the linear case, and it will provide great insight for nonlinear problems. |
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May 7 |
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Second-order nonlinear ODE with Dirac Delta True. I've edited accordingly. |
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May 6 |
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Parabolic PDEs: Boundary conditions Well, $B_a[u(x,t)] = T(t) B_a[X]$ right? Now $T(t)B_a[X] = 0$ for all $t$, hence $B_a[X] = 0$. |
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Mar 7 |
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The number of non-trivial polynomial solutions of the differential equation $x^3y'(x)=y(x^2)$ @user33640 According to Wookipedia, "anzats" were a dangerous and mysterious Force-sensitive near-Human species with two tentacle-like proboscises that curled out and extended from their cheeks, with which the Anzati were able to feed upon the brains of their prey. An ansatz is a proposed form for the solution of an equation :) |
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Feb 22 |
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To get addition formula of $\tan (x)$ via analytic methods @Mathlover I added other examples. The method works with Euler equations $$\frac{dx}{\sqrt{X}} + \frac{dy}{\sqrt{Y}} = 0,$$ where \begin{align}{}X &= a_0x^4 + a_1 x^3 + a_2 x^2 + a_3 x + a_4\\ Y &= a_0y^4 + a_1 y^3 + a_2 y^2 + a_3 y + a_4\end{align} It strongly relies on the construction of the second solution. For the problem $y' = 1 + y^n$, this doesn't seem to be straight forward, but maybe some modification can be attempted. |
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Feb 19 |
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To get addition formula of $\tan (x)$ via analytic methods Do we know $\tan(x) = \frac{\sin(x)}{\cos(x)}$, $\sin^2(x) + \cos^2(x) = 1$ and $\cos(x+y) = \cos(x)\cos(y) - \sin(x)\sin(y)$? |
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Feb 12 |
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PDE method of characteristics solving $e^{t^2}u_t+tu_x=0$ with $u(x,0)=x+2$ @user61933 I've edited the answer in order to address your questions. Please, if you find it appropriate, vote up and accept the answer. |
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Feb 12 |
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PDE method of characteristics solving $e^{t^2}u_t+tu_x=0$ with $u(x,0)=x+2$ I added $\LaTeX$ to your question. Please verify it's correct. |
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Jan 30 |
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Diff eq. transformation polar coordinates Two things. Your notation is obscure, as $t$ is function of the variable of integration, i.e. $\frac{d t}{d \xi} \equiv {t}'$, hence you are using the chain rule wrong; also, the cubic term on the second component of your substitution should read $-r^3 \sin t$. See my answer for details. |
