Consider the partial differential equation
$\frac{\partial^2 u}{\partial x^2} = \frac{\partial u}{\partial t}$
with boundary conditions $u(0, t) = 0$ and $u(\frac{1}{4}\pi, t) = 0$ for $t \ge 0$, and the initial condition
$u(x, 0) = x(-x + \frac{1}{4}\pi)$.
The general solution that satisfies the boundary conditions is
$u(x, t) = \sum_{n=1}^{\infty} A_n e^{-16n^2t} \sin(4nx)$.
Select the option that gives the correct method for determining the constants $A_n$ from the given initial
condition.
Select one:
$\circ \quad A_n = \frac{8}{\pi} \int_0^{\frac{1}{4}\pi} x(-x + \frac{1}{4}\pi) \sin(4nx) dx$
$\circ \quad A_n = \frac{8}{\pi} \int_0^{\frac{1}{4}\pi} x(-x + \frac{1}{4}\pi) \cos(4nx) dx$
$\circ \quad A_n = \frac{4}{\pi} \int_0^{\frac{1}{4}\pi} x(-x + \frac{1}{4}\pi) \sin(4nx) dx$
$\circ \quad A_n = \frac{4}{\pi} \int_0^{\frac{1}{4}\pi} x(-x + \frac{1}{4}\pi) \cos(4nx) dx$
$\circ \quad A_n = \frac{8}{\pi} \int_0^{\frac{1}{4}\pi} x(-x + \frac{1}{4}\pi) \sin(4nx) dx$
