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\begin{document}
{\LARGE
\begin{center}
MATH 220 E1 ~~~~~~ QUIZ \#14(Last)  ~~~~~  (20 points)
\end{center}
(12 minutes, 23 April 1999 \{14th Week\}, PDEs: 10.5)
\\*[0.2em]
NAME:
\\*[0.1em]
SOCIAL SECURITY NUMBER (SSN):
\\*[0.1em]
DISCUSSION TIME AND DAY (DT\&D):}
\\*[0.1em]
{\B 1. Formally Solve IVP/BVP for Heat Equation:}
\EQ{
\mathrm{PDE:}~~
\frac{\partial u}{\partial t}
&=& \frac{\partial^2 u}{\partial x^2} 
- 6\cdot x\cdot(\pi-2\cdot x)~,
\\*[0.3em]
&&
~~ 0<x<\pi/2~, ~ t>0~;\\*[-2.6em]
}
\EQ{
\mathrm{BCs:}~~
u(0,t) = 0, &&~~ u(\pi/2,t) = 50\pi +(\pi/2)^4, 
\\*[0.3em]&&~~ t>0~;\\*[-2.6em]
}
\EQ{
\mathrm{IC:}~~
u(x,0)=100\cdot x + x^3\cdot(\pi-x) \\*[0.3em]
+ 64\cdot\sin(2\cdot x)
~~ 0 <x< \pi / 2~;
}
{\B by Separation of Variables and Nonhomogeneous Terms, with
Fourier Series Methods.
\underline{\B You Must Show Your Work.}}
\\*[1.5em]
{\I\Large Fourier Sine Coefficient for $f(x)$ on $0<x<L$ (if needed):}
\eq{
\mathrm{FSC:}~~
b_n=\frac{2}{L}\int_0^L
f(x)\sin\left(\frac{n\pi x}{L}\right) dx ;
}
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