DEPARTMENT OF MATHEMATICS & STATISTICS

MATH 2503

FINAL EXAMINATION
December 1997
TIME: 3 HOURS

NO CALCULATORS

MARKS
(8) 1. Solve the following initial value problems:
(a) $y' = e^{-y} \sin x,\;\;\;y(0) = 1$;
(b) $xy' + 2y = 2x,\;\;\;\; (x > 0),\;\;\;\;y(1) = 3$.
(13)2. Find the general solution to the following equations:
(a) y'' - 2y' - 3y = 2e3x;
(b) ${\displaystyle \frac{d^2y}{dx^2} +
4y = 3 \cos x}$;
(c) y'' + 2y' = x + 2.
(7)3. Use variation of parameters to find the general solution of
\begin{displaymath}y'' - 2y' + y = \frac{e^x}{x}.
\end{displaymath}
(7)4. Use power series to solve the following differential equation. Find the recurrence relation and terms up to and including those involving x5 in the general solution of
y'' + 2xy' + 2y = 0.
(16)5. Test the following series for convergence or divergence:
(a) ${\displaystyle \sum_{n=1}^{\infty}\;\frac{n+1}{n+100}}$ (b) ${\displaystyle \sum_{n=2}\;\frac{1}{n(\ln
n)^2}}$
(c) ${\displaystyle \sum_{n=1}^{\infty}\;\frac{n+1}{\sqrt{4n^3
+ 1}}}$ (d) ${\displaystyle \sum_{n=1}^{\infty}\;3^{2n}
5^{1-3n}}$
(17)6. (a) Determine whether the series ${\displaystyle
\sum_{n=1}^{\infty}\;\frac{(-1)^n \sqrt{n}}{n+1}}$ is absolutely convergent, conditionally convergent or divergent. Give reasons.
(b) Find the interval of convergence of the series
\begin{displaymath}\sum_{n=0}^{\infty}\;\frac{(-2)^nx^n}{\sqrt{n+1}}.
\end{displaymath}
(c) Give the first four non-zero terms in the expansion of
\begin{displaymath}\frac{1}{\sqrt{4 - x}}.
\end{displaymath}
(10)7. Use Gauss-Jordan elimination method to solve
${\displaystyle \begin{array}{rrrrrrcr}
\mbox{(a)} & 2x & - & 4y & + & 6z & = & ...
...& 4x & - & 3y & + & z & = & 8\\
& 2x & + & 5y & - & 3z & = & -6.
\end{array}}$
${\displaystyle \begin{array}{rrrrrrrrcr}
\mbox{(b)} & x & & & + & 7z & - & 2w &...
... + & 11z & - & 12w & = & -7\\
& & & y & - & 2z & + & 5w & = & 4.
\end{array}}$
(11)8. (a) Find the inverse of the matrix
\begin{displaymath}A = \left( \begin{array}{rrr}
1 & 1 & -1\\
0 & 1 & 3\\
2 & 1 & -4
\end{array} \right) \end{displaymath}
and use this to find the solution of the system
\begin{displaymath}\begin{array}{rcrcrcr}
x & + & y & - & z & = & 2\\
& & y & + & 3z & = & -1\\
2x & + & y & - & 4z & = & 3
\end{array} . \end{displaymath}
(b) Use Cramer's rule to solve the system
\begin{displaymath}\begin{array}{rcrcrcr}
x & + & y & - & z & = & 0\\
2x & + & 2y & + & z & = & 0\\
& & y & - & 5z & = & 1
\end{array} \end{displaymath}
for x.
(9)9. (a) Show that the vector ${\displaystyle \bar{x} =
\left( \begin{array}{r}
3\\ -6\\ 2
\end{array} \right)}$ is an eigenvector of
\begin{displaymath}M = \left( \begin{array}{rrr}
-5 & -5 & -9\\
8 & 9 & 18\\
-2 & -3 & -7
\end{array} \right). \end{displaymath}
(b) (i) Find the characteristic equation of
\begin{displaymath}A = \left( \begin{array}{rrr}
4 & 0 & 1\\
-2 & 1 & 0\\
-2 & 0 & 1
\end{array} \right) . \end{displaymath}
(ii) Show $\lambda = 2$ is an eigenvalue of A and find the eigenvectors corresponding to it.

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