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{\bf Name:}  \vrule height .4pt depth 0 pt width 2.0
in  \hskip .5 in {\bf Professor:} A.Alexandrou Himonas
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 \centerline{\bf {Mathematics 226}}
\centerline{\bf {Differential Equations and Linear Algebra}}
\centerline{\bf Fall Semester 1991}
\centerline{\bf Exam 1}
\centerline{\bf {September 23, 1991}} 

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This Examination contains five problems worth a total of  100  points, 
each problem worth 20 points, on (7) sheets of paper including the
front cover and one extra sheet on the back.  Do all your work in this
booklet and show your computations.  Calculators, books and notes are
not allowed.
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{\bf Sign the pledge:}  \lq \lq {On my honor, 
I have neither given nor received unauthorized aid on this Exam}":
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\centerline{\bf GOOD LUCK}


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\noindent
1.  Solve the DE's
$$\frac{dy}{dx}=e^{x+y}\tag1.1$$
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$$x\frac{dy}{dx}=y+\frac {x}{cos(y/x)}\tag 1.2$$
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2. Solve the following initial value problem:
$$\frac{dy}{dx}+\frac {1}{100+x}y=4,\ y(0)=100.$$

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3. Find $a$ for which the following  DE is exact and then solve it.
$$(y-3x^2+4)+(ax+4y^3-2y)\frac{dy}{dx}=0.$$

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4. Assume that an ice cube(its volume) is melting at a rate proportional to its
surface area. If its edge at the begining is 6 cm, and  20 minutes later is 4 cm, find
the length of its edge at any time t. When will the the ice melt completely?

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5. Consider a tank holding  400  gallons of a salt solution with 
concentration  0.25 lb of salt per gallon.  A solution containing  0.5 
lb of salt per gallon is pumped into the tank at the rate of  8  gal/min, and the
well-stirred mixture flows out of the tank at the rate of  4  gal/min. 
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{\bf a.}	Write the initial value problem (i.e. a  DE  and an initial condition) 
needed to find the amount  $S(t)$ of salt in the tank at any time  $t$.  
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{\bf b.}   Now solve this initial value problem to find $S(t)$ (be careful you may
have  already solved it !). 

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