# Trying to solve a partial differential equation using d'Alembert's solution

1. Mar 11, 2009

### jaejoon89

Hi, I'm trying to understand this.

The given equation is y_tt = 4 y_xx
0 < x < pi, t>0
where y_tt is the 2nd derivative with respect to t, y_xx is 2nd wrt x

Boundary conditions
y(0,t) = 0 and y(pi,t) = 0

And initial conditions
y_t (x,0) = 0 = g(x)
y(x,0) = sin^2 x = f(x)

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My teacher wrote that F(x) is the odd periodic extension of f(x), and then wrote

F(x) = sign(sinx)sin^2 x

1) I assume this is to make it odd but why wouldn't he just write sign(x)sin^2 x?

2) Also, there was a similar question in class but -infinity < x < infinity and no boundary conditions given with one of the initial conditions y(x,0) = 1/(1+x^2) = f(x). In that case, since f(x) is even why isn't it necessary to use the sign function?

2. Mar 11, 2009

### Hurkyl

Staff Emeritus
Oddness wasn't the only property he wanted.

3. Mar 11, 2009

### jaejoon89

Why does the sin^2 need to be corrected like that?