DEPARTMENT OF MATHEMATICS & STATISTICS

MATH 2503

FINAL EXAMINATION
April 1998
TIME: 3 HOURS

NO CALCULATORS. SHOW ALL CALCULATIONS.

MARKS
(8) 1. Solve the following initial value problems:
(a) $(x-1)(y+1)dx - xy\;dy = 0;\;\;\;\;y(1) = 0$;
(b) ${\displaystyle \frac{dy}{dx} + y \tan x = \cos x,\;\;y(0) =
2}$.
(12)2. Find the general solution to the following equations:
(a) y'' - 2y' + y = x2 + 1;
(b) y'' - 3y' + 2y = ex;
(c) $y'' + y = -3 \sin 2x$.
(8)3. Use variation of parameters to find the general solution of
\begin{displaymath}y'' - 6y' + 9y = \frac{e^{3x}}{x^2}\;\;.
\end{displaymath}
(8)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.
(8)5. Find the sum of the following series:
(a) ${\displaystyle \sum_{n=1}^{\infty}\;\frac{1}{n^2 + 3n + 2}}$ (b) ${\displaystyle \sum_{n=1}^{\infty}\;3^{2n}
5^{1-3n}}$.
(12)6. Test the following series for convergence or divergence:
(a) ${\displaystyle \sum_{n=1}^{\infty}\;\frac{2n^2+n+1}{3n^2-n+4}}$ (b) ${\displaystyle \sum_{n=1}^{\infty}\;\frac{\ln
n}{n}}$ (c) ${\displaystyle
\sum_{n=1}^{\infty}\;\frac{n^2+n+1}{\sqrt{n^7 + n^5 + 1}}}$.
(12)7. (a) Find the interval of convergence of the series
\begin{displaymath}\sum_{n=1}^{\infty}\;\frac{(-1)^n(x+2)^n}{3^n \sqrt{n^2+1}}.
\end{displaymath}
(b) Find the Taylor series for f(x) = e2x about a = 1. Write the general term.
(8)8. Use Gauss-Jordan elimination method to solve
\begin{displaymath}\left. \begin{array}{rcrcrcr}
x & - & y & + & z & = & 4\\
x ...
... & 3\\
3x & - & y & + & 2z & = & 11
\end{array} \right. \;\;. \end{displaymath}
(14)9. (a) Find the inverse of the matrix
\begin{displaymath}A = \left[ \begin{array}{rrr}
1 & 0 & 1\\
2 & 1 & 0\\
0 & 1 & -1
\end{array} \right] \end{displaymath}
and use this to find the solution of the system
\begin{displaymath}\left. \begin{array}{rcrcrcr}
x & & & + & z & = & 1\\
2x & ...
...& & & = & 2\\
& & y & - & z & = & 3
\end{array} \right.\;\;. \end{displaymath}
(b) Use Cramer's rule to solve the system
\begin{displaymath}\left. \begin{array}{rcrcrcl}
x & + & 2y & + & 3z & = & 0\\
...
... + & 2z & = & 1\\
& & y & + & 2z & = & 2
\end{array} \right. \end{displaymath}
for y.
(10)10. (a) Find the characteristic equation of
\begin{displaymath}A = \left[ \begin{array}{rrr}
1 & -1 & 4\\
3 & 2 & -1\\
2 & 1 & -1
\end{array} \right]\;\;. \end{displaymath}
(b) Show $\lambda = 3$ is an eigenvalue of A.
(c) Find the eigenvectors of A corresponding to the smallest eigenvalue of A.

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