Help with a differential equation

[ElvenVeil]

Homework Statement

Hello

I am new to this forum, but I hope I can get help with a problem I haven't been able to figure out what to do with.

info:

we have a one dimensional equation -d/dx [a(x) du/dx] = p(x)

where we seek a solution u(x) where x is within [0,1] , that satisfies the 2 boundary conditions u(0) = 0, u(1) = 0

p and a is given by a(x) = 1 and p(x) = 1

any help with this problem would be very nice. On beforehand thanks

Homework Equations

The Attempt at a Solution

My thought was to first find a general solution to the equation and then insert the conditions given. I have a hard time finding a general solution to the equation (-d/dx [a(x) du/dx] = p(x) ) so I feel a little stuck on how to approach it.

 

[Fightfish]

p and a is given by a(x) = 1 and p(x) = 1

Erm...just use this? I don't think it is possible to obtain a closed form expression for the general solution for arbitrary a(x) and p(x).

 

[HallsofIvy]

The general solution is very difficult- although I think it can be done in terms of
$$u(x)= -\int_x^1\frac{1}{a(u)}\int_0^u p(t)dt du$$

But you are give that a(x)= 1 and p(x)= 1 so the problem is to solve $\frac{d^2x}{dt^2}= -1$ which can be done by two simple integrations.