# Some Differential Equation help needed

1. Sep 20, 2005

### stunner5000pt

Of the Partial Kind
Using d'Alemberts soltuion for the vibrating string in one dimension

Find u(1/2,3/2), when l-=1, c=1, f(x) = 0, g(x) = x(1-x)
Now i tried simply substituting this into the solution that is (since f(x)=0)
$$u(x,t) = \frac{1}{2} \int_{x-t}^{x+t} g(x) dx$$
but it yields the wrong answer.
Does the length of the string have anything to do with the answer?

2. Sep 21, 2005

### saltydog

Stunner, to use D'Alembert's forumua, you need to remember to use the "odd extensions" of both f(x) and g(x). Now, I know that's not pretty but that's just how it is. Remember when I said that $Sin[\pi x]$ was already an odd-extension and so we didn't have to do anything about it? That's not the case with g(x)=x(1-x) over the interval you're integrating from. Look at the first plot. That's g(x) un-extended. We wish to make an odd periodic function of g(x) over the interval of integration. In your case thats 1/2-3/2 to 1/2+3/2 or the interval [-1,2]. So, first thing is to "odd-extend" what the function looks like in [0,1] to the interval [-1,0]. Well, that's the second plot and the equation for it is:

$$g_1[x]=x(1+x)$$

The equation for the interval [0,1] is just g(x):

$$g_2(x)=x(1-x)$$

Now I wish to do that again for the interval [1,2], that is an odd extension of g(x) which would just be flipping it over into the interval [1,2]. The equation for that one would be:

$$g_3(x)=-(x-1)+(x-1)^2$$

The third plot is all three. So:

\begin{align*} u(1/2,3/2) &=\frac{1}{2} \int_{-1}^{2}\tilde{g_0}(\tau)d\tau \\ &= \frac{1}{2}\left(\int_{-1}^0 g_1(\tau)d\tau+\int_0^1 g_2(\tau)d\tau+\int_1^2 g_3(\tau)\tau \right) \end{align}

I get -1/12. Is that what you get?

Edit: Stunner, I initially made a typo on g3 but corrected it above.

Edit2: Forgot the 1/2 in front of the integral sign. Suppose that's -1/12 now. Sorry.

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Last edited: Sep 21, 2005
3. Sep 21, 2005

### stunner5000pt

it is quite clear that we are not being taught (or the material's presentation) correctly. I did not know how to extend the functions. I understand now... thank you very much!