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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 }
\centerline{\bf Final Exam}
\centerline{\bf {December 20, 1991, 8:00-10:00 AM}} 

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This Examination contains eight problems worth a total of  150  points, 
each problem worth 20 points,except the last which worths 10 points, on 10 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 and HAPPY HOLIDAYS}

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1.  (a)  Solve the initial value problem $y' - xy + 2x = 0, y(0) = 0.$
 

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(b)  Solve the DE $(xe^{xy} + 4y^3) \frac{dy}{dx} = 2x - ye^{xy}$

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2.  Given that $y_1 = e^x$ and $y_2 = x^2e^x$ are two solutions to the DE
$$xy'' - (2x + 1)y^1 + (x + 1)y = 0,$$
find the general solution to the DE $xy'' - (2x + 1)y' + (x + 1)y = x(x^2 - 2x -1)e^x.$

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3.  Let $y(x) = \sum_{n=0}^{\infty} a_n (x-3)^n$ be the power series expansion of the
solution to the initial value problem $y'' - 2(x-3)y' - 2y =0, y(3) =0, y'(3) = 1.$  \newline

a)  Find the recurrence relation for the coefficients $a_n.$

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b)  Then compute all $a_n$ and write the solution $y(x).$  What is its radius of
convergence?


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4.  Let $$A=\pmatrix  1&3&-5&2&-7\\
-4&-9&17&-8&34\\
-3&6&12&-5&24\\2&3&-7&4&-8
\endpmatrix , x = \pmatrix x_1\\x_2\\x_3\\x_4\\x_5\endpmatrix$$

a)  Find the reduced echelon form of $A.$

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b)  Find the solution space $W$ of the system $Ax = 0$ and a basis for $W.$

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c)  Find the rank of $A.$  Then find $dim(W).$

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5.  
a)  By row elementary operations $A$ are reduced to
$$\pmatrix 1&3&-5&2&-7\\
0&3&-3&0&6\\
0&3&-3&1&3\\
0&-3&3&0&6
\endpmatrix     \to  \pmatrix 1&3&-5&2&-7\\
0&1&-1&0&2\\0&0&0&1&-3\\0&0&0&0&0 \endpmatrix$$

$$\pmatrix 1&3&-5&0&1\\
0&1&-1&0&2\\
0&0&0&1&-3\\
0&0&0&0&0 \endpmatrix     \to  \pmatrix 1&0&-2&0&-5\\
0&1&-1&0&2\\
0&0&0&1&0-3\\
0&0&0&0&0 \endpmatrix$$

b)  The system $Ax = 0$ is reduced to
$$\aligned
x_1 \hphantom{x_2} -2x_3 \hphantom{x_4} -5x_5 &=0\\
\hphantom{x_1}x_2 -x_3 \hphantom{x_4}+2x_5&=0\\ 
\hphantom{x_1} \hphantom{x_2} \hphantom{x_3}x_4 -3x_5&=0 \endaligned$$

Let $x_5 = \alpha,$   $x_3 =\beta$  then $x_4 = 3\alpha,$ $x_2 =\beta - 2 \alpha$
and $x_1 = 5 \alpha + 2 \beta$ and

$$W =  \left\{ 
\pmatrix 
x_1\\
x_2\\
x_3\\
x_4\\
x_5 \endpmatrix = \pmatrix 5 \alpha + 2 \beta\\
\beta - 2 \alpha\\
3 \alpha\\
\beta\\
\alpha \endpmatrix = \alpha \pmatrix 5\\-2\\3\\0\\1 \endpmatrix + \beta \pmatrix
2\\1\\0\\1\\0 \endpmatrix : \alpha, \beta, \Bbb R \right\}$$

A basis for $W$ is $\left\{ \pmatrix 5\\ -2\\ 3\\ 0\\ 1 \endpmatrix , \pmatrix
2\\ 1\\ 0\\ 1\\ 0 \endpmatrix \right\}$

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5.  Use Cramer's Rule to solve the system: \newline

$$\aligned
3x_1 + x_2 \hphantom{x_3} &= 1\\
-2x_1 -4x_2+3x_3 &= -7\\
5x_1 + 4x_2 -2x_3 &=6 \endaligned$$

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6.  Find the general solution of the system of DE
$$\frac{dx}{dt} = \pmatrix 1 & 2\\ 4 & 3 \endpmatrix x,  \;  x = \pmatrix x_1(t)\\ x_2(t)
\endpmatrix$$

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7.  Assume that the human population $p = p(t)$ of the earth obeys the logistic equation
$$\frac{dp}{dt} = r(1 - \frac{P}{K})P, \;  r = 0.029,  \; K = 10^{10}.$$

If the population now is $\frac{1}{2}10^{10}$, find the population $P(t)$ at any time
$t.$  \newline
Also find $\displaystyle \lim_{t \to \infty} P(t).$
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8.  Find all terms up to degree four of the Taylor Series expansion $a + x_0 = 0$ of the
function $y = y(x)$, where $y$ is the solution to the following initial value problem

$$y'' - xy' - e^{2y} = 0, \; \; y(0) = 0, \;\; y'(0) = 1.$$

  
 

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