# Tagged Questions

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### Lengendre Polynomial calculation

I'm just getting learning about how legendre polynomials come about when considering product solutions in spherical coordinates with azimuthal symmetry. I'm trying a problem on my own, and I'm a bit ...
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### Heat equation: Why are these ratios of functions constant

One can solve the heat equation $$\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2}$$ by a separation of variables such that $u(x, t) = f(x)g(t)$. Substituting this into the ...
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### System of PDE's with unknown functions

So by messing around with some stuff in my own research I came across this problem and I have no idea how to proceed. I suspect it may have something to do with solving systems of PDE's but I could be ...
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### Integration by parts on all of $\mathbb{R}^n$ with $n>1$

So this came up as I was thinking about the uniqueness of solutions to the wave equation. I have seen proofs for uniqueness on all of $\mathbb{R}$ or on bounded subsets of $\mathbb{R}^n$, but never ...
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### a simple calculation

Can anyone see how (1) lead to (2)? \begin{align} ...
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### Partial differential equation, mixed derivatives

What can be concluded from following equation: $$\frac{\partial f(x,y)}{\partial x}-\frac{\partial g(x,y)}{\partial y} = 0$$ where $f(x,y)$ and $g(x,y)$ are functions of two independent variables ...
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### Evans 's PDE proof

Again, I got stuck. Please help me to understand the following: What is the meaning when you change from integration over the Ball B(x,r) to the surface integration dB(x,s), with another integral ...
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### Solving PDE by Canonical form transformation

For reference, the entire equation to be solved for $u(x,y)$ is: $A= -2x^2-8xy-8y^2+42x-14y$ $B= -5x^2-20xy-20y^2+105x-35y$ $C= 3x^2+12xy+12y^2-63x+21y$ $E= 28x+56y$ $K= -14x-28y$ where ...
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### Find vector field given curl

I have an equation $\nabla \times \vec{B} = \mu_{0}\vec{J}$, where $\vec{J} = \left\langle f(x,y), g(x,y), 0 \right\rangle$ and need to solve for $\vec{B}$. I've looked elsewhere on here for how to ...
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### What is the order of the PDE $\newcommand\pp\partial\frac{\pp^2u}{\pp x^2}+\frac{\pp^3u}{\pp x^2 \pp y}+\frac{\pp^2u}{\pp^2y}=xy\frac{\pp u}{\pp x}$? [closed]

The order of the differential equation $$\frac{\partial^2 u}{\partial x^2}+\frac{\partial^3 u}{\partial x^2 \partial y}+\frac{\partial^2 u}{\partial^2 y}=xy\frac{\partial u}{\partial x}$$ is ...
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### Solving the Telegraph Equation using Partial Differential Equations and Sturm-Liouville theory

I've been asked to do the following question, and I've got through the brunt of it (so this is going to be a rather long question...), but I'm just having a bit of trouble applying Sturm-Liouville ...
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### Finding the partial derivatives of $V (x, y) = U (x, y)e^{−ax−by}$

I think I did something wrong, so I was hoping someone might be able to show me the solution Two functions $V (x, y)$ and $U (x, y)$ are connected by the equation $$V (x, y) = U (x, y)e^{−ax−by}$$ ...
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### Finding the change of variables to transform $u_{tt} - u_{xx} = 0$ into $u_{rs} = 0$

I'm just beginning to introduce myself to partial differential equations and one of the first problems presented in the textbook I have literally no idea how to do. I think the author intended the ...
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### Does every function with $f_x,f_y>0,f_{xx},f_{yy}<0$ with particular condition have to satisfy $f_{xy}/f_{xx} = -x/y$?

Let continuous real functions $f$ of two real variables $x,y$ satisfy the following condition: (Let us define $f_{xx}:=\frac{\partial^2 f}{\partial x^2}$ and $f_x:=\frac{\partial f}{\partial x}$, and ...
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### Clarifying definition of outward unit normal

I would like to figure out how to properly define the outward unit normal vector $\nu$ for a bounded domain $\Omega \subset \mathbb{R}^n$ with smooth boundary $\partial \Omega$ ($n \ge 2$). I am ...