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{\large

\begin{center}
{\bf  MATH 220 E1 \hspace{0.4in} EXAM 1 \hspace{0.4in} 
12pm 26 February 1999 \hspace{0.4in} Hanson}
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{\bf Directions:}  Answer {\bf All Questions} and show {\bf All Work} in the
{\bf Exam Booklet} provided.  Write your {\bf Name}, {\bf Social Security
Number}, and {\bf Discussion Section Hour/Day} on the Exam Book Cover Page.
Keep your eyes on your own work and keep your own work covered.
\\{\centering\underline{~\hspace*{6.85in}~}}
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\begin{enumerate}
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\item A large tank initially has {\bf 1.5~L} (Liters) of pure water.  
At time $\DS t=0$, a sugar solution enters the tank at {\bf 5.5~L/Min} 
with sugar concentration of {\bf 2~KG/L}.  
The well-stirred solution leaves the tank at the rate of {\bf 2.0~L/Min}.  
{\it State} and {\it solve} for $\DS A(t) =$ Amount of sugar in 
the tank at time $\DS t$. 
\hfill(20 points)
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\item Solve the Initial Value Problem explicitly for $\DS y(x)$:
$$\DS \frac{\DS dy}{\DS dx} = x e^{\DS y} + e^{\DS 2x+y},~~~~~~~ y(0) = 0.$$ 
{\it\{Hint: Obey the law of exponents!\}} 
\hfill(20 points)
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\item
\begin{enumerate}
\item First find two linearly independent solutions to the 
homogeneous ODE corresponding to 
$$\DS y^{\DS\prime\prime}(x) + 9y(x) = 7 - 5e^{\DS 3x} + 6cos(3x),$$
and show that they are linearly independent using the Wronskian.
\item  Next find the general solution of above inhomogeneous ODE.\\
{\it\{Hint: How about method of underdetermined coefficients?\}} 
\hfill(20 points)
\end{enumerate}
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\item  
\begin{enumerate}
\item First show that $y_h(x)=x^3$ is the solution for the 
homogeneous ODE corresponding to 
$$\DS x^2y^{\DS\prime\prime}(x) - 5xy^{\DS\prime}(x)+ 9y(x) = 2x^3 ~;$$
\item Find the general solution of the above inhomogeneous ODE.
\hfill(20 points)
\end{enumerate}
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\item  
\begin{enumerate}
\item First find the Euler's method approximation at $\DS x = 1.05$
using a $\DS h = 0.05$ step to the solution of the IVP,
$$\DS y^{\DS\prime}(x) = -x^2y(x) - 3y^2(x),~~~~~~~ y(1) = 2~;$$ 
\item Next find the Improved Euler Correction at $\DS x = 1.05$.\\
{\it\{Hint: The correction uses Euler's Method as the predictor step
in an approximation to the trapezoidal rule!\}} 
\hfill(20 points)\\
\end{enumerate}
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\end{enumerate}
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