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Question: Solve Laplace’s Equation

Sunday, 9 of March , 2008 at 1:10 pm

Question: Find the solutions to Laplace’s Equation:

\frac{\partial^2 u(x,y)}{\partial x^2} + \frac{\partial^2 u(x,y)}{\partial y^2} = 0

Note: Since there are no boundary conditions, there won’t be an exact solution.

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Category: Differential Equations, Questions

Answer: Falling Infinite Rope

Friday, 7 of March , 2008 at 8:21 pm

Question: An infinite rope with a linear density of \lambda is placed on a frictionless table. If the end of the rope is placed at the end of the table and starts falling, what is the velocity of the rope as a function of distance?

Answer: According to Newton’s 2nd Law of Motion, we know that

F_{net} = \frac{\partial p}{\partial t}

If we create a force diagram, we can easily see that F_{net} = mg.

where m = \lambda x.

Therefore, we can create the equation of motion as follows:

\frac{\partial p}{\partial t} = \frac{\partial (m \dot{x})}{\partial t} = \dot{x}\dot{m} + m\ddot{x} = mg
\dot{x}\frac{\partial (\lambda x)}{\partial t} + \lambda x \ddot{x} = \lambda x g
\lambda\dot{x}^2 + \lambda x \ddot{x} = \lambda x g
\dot{x}^2 + x \ddot{x} = x g

In order to solve this differential equation, let y = \frac{1}{2} \dot{x}^2.

This means:

\dot{y} = \dot{x} \ddot{x} ~~~\Longrightarrow ~~~ \ddot{x} = \frac{\dot{y}}{\dot{x}} = \frac{\frac{\partial y}{\partial t}}{\frac{\partial x}{\partial t}} = \frac{\partial y}{\partial x}

Therefore, the differential equation becomes

2y + x \frac{dy}{dx} = gx ~~~\Longrightarrow ~~~ \frac{dy}{dx} + \frac{2}{x}y = g

We can solve this using the integrating factor method. According to our differential equation, our integrating factor will be e^{\int \frac{2 dx}{x}}= x^2. If we multiple our integrating factor to our ODE, we get

x^2 \frac{dy}{dx} + 2xy = g x^2
\frac{d(x^2 y)}{dx} = gx^2
x^2 y = \int g x^2 dx = \frac{gx^2}{3} + C
\therefore y = \frac{g}{3}x + \frac{C}{x^2}

However, since y = \frac{1}{2} \dot{x}^2, we know that

\frac{1}{2}\dot{x}^2 = \frac{g}{3}x + \frac{C}{x^2}
\dot{x} = \sqrt{\frac{2}{3}gx + \frac{2C}{x^2}}

If we add the boundary condition, \dot{x}(0) =0, the equation reduces to

\dot{x} = \sqrt{\frac{2}{3}gx}

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Category: Answers, Classical Mechanics

Question: Falling Infinite Rope

Sunday, 2 of March , 2008 at 9:42 pm

Question: An infinite rope with a linear density of \lambda is placed on a frictionless table. If the end of the rope is placed at the end of the table and starts falling, what is the velocity of the rope as a function of distance?

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Category: Classical Mechanics, Questions

Answer: General Form of Sequence

Saturday, 1 of March , 2008 at 9:59 pm

Question: The solution to x = \sqrt{10y+1} where x,y \in \mathbb N yields the following sequence:

8,~12,~36,~44,~84,~96,~152,~168,~240~\ldots

Find the general form for this sequence.

Answer: When analyzing this sequence, the first thing I tried was comparing the difference between each number in the sequence.

Lets investigate this new sequence.

4,~24,~8,~40,~12,~56,~16,~72~\ldots

It appears the odd terms are related to each other and so are the even terms. Lets break the odd terms and even terms into two different sequences.

odd~terms:~4,~8,~12,~16,\ldots
even~terms:~24,~40,~56,~72,\ldots

As you can see, the odd terms has the form:

\phi_n = 4n

The even terms, likewise, can be seen to have the form:

\phi_n = \phi_{n-1} + 16 or
\phi_n = 16n + 8

If we merge these results, we can get pattern for the following sequence

4,~24,~8,~40,~12,~56,~16,~72~\ldots

which is

\phi_n = \begin{cases}4(\frac{n+1}{2}) & odd~n \\ 16(\frac{n}{2}) +8 & even~n\end{cases}
\phi_n = \begin{cases}2(n+1) & odd~n \\ 8(n+1) & even~n\end{cases}

We can then use this result to find the solution to

8,~12,~36,~44,~84,~96,~152,~168,~240~\ldots

which would be

\psi_n = \psi_{n-1} + \phi_n
\psi_n= \psi_{n-1} + \begin{cases}2(n+1) & odd~n \\ 8(n+1) & even~n\end{cases}

where \psi_0 = 8.

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Category: Algebra, Answers

Question: General Form of Sequence

Sunday, 24 of February , 2008 at 12:34 pm

Question: The solution to x = \sqrt{10y+1} where x,y \in \mathbb N yields the following sequence:

8, 12, 36, 44, 84, 96, 152, 168, 240…

Find the general form for this sequence.

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Category: Algebra, Questions