DEPARTMENT OF MATHEMATICS & STATISTICS

MATH 2503

FINAL EXAMINATION
APRIL 1999
TIME: 3 HOURS

NO CALCULATORS.
SHOW ALL INTERMEDIATE CALCULATIONS.

MARKS
(5) 1. Solve the initial value problem:
\begin{displaymath}t\;\frac{dy}{dt} = (y-2)(1-3t)\;,\;\;\;\;y(1) = 3.\end{displaymath}

Find ${\displaystyle \lim_{t \rightarrow \infty} y(t)}$.

(5) 2. Find the general solution to the differential equation:
y'' + 2y' + y = 4e-x.
(8) 3. Use the method of variation of parameters to solve
y'' + 3y' + 2y = cos(ex).
(3) 4. Find ${\displaystyle \sum_{n=1}^{\infty}\;
\frac{3^{\textstyle n}}{4^{\textstyle n+2}}}$, if the series converges.
(6) 5. Test the following series for convergence:
(a) ${\displaystyle \sum_{n=1}^{\infty}\;\left( 2 + \frac{3}{n}
\right)^{\textstyle n}}$; (b) ${\displaystyle
\sum_{n=1}^{\infty}\;\frac{n}{e^{\textstyle n/2}}}$.
(5) 6. Determine whether ${\displaystyle
\sum_{n=1}^{\infty}\;\frac{(-1)^{\textstyle
n+1}(3n+2)}{\sqrt{n}(3n-5)}}$ is absolutely convergent, conditionally convergent or divergent.
(7) 7. Find the interval of convergence for the power series
\begin{displaymath}\sum_{n=1}^{\infty}\;\frac{2^{\textstyle n} (x-3)^{\textstyle
n}}{n(n+1)}\;\;. \end{displaymath}
(4) 8. To what function does ${\displaystyle
\sum_{n=1}^{\infty}\;\frac{x^{\textstyle n}}{n}}$ converge when it converges?
(5) 9. Find the Taylor series for xex about x = 2.
(8) 10. Use a power series to find the solution of the following initial value problem:
y'' - x2y' - 2xy = 0, y(0) = 1, y'(0) = 0.
(5) 11. Use Gauss-Jordan elimination to find all solutions or show that there are none:
\begin{displaymath}\left. \begin{array}{rcrcrcrcr}
2x_1 & - & x_2 & + & 2x_3 & -...
... & -10\\
& & x_2 & & & & 3x_4 & = & 12
\end{array} \right. . \end{displaymath}
(7) 12. (a) Find the inverse of the matrix
\begin{displaymath}A = \left[ \begin{array}{rrr}
1 & 2 & 3\\
3 & -1 & 0\\
-1 & 1 & 1
\end{array} \right] . \end{displaymath}
(b) Use the inverse found in part (a) to solve the system:
\begin{displaymath}\left. \begin{array}{rcrcrcr}
x & + & 2y & + & 3z & = & 5\\ ...
...& & = & 2\\
-x & + & y & + & z & = & -2
\end{array} \right. . \end{displaymath}
(4) 13. If A is an n × n matrix with the property A-1 = 2A determine the value(s) of detA. Give a 2 × 2 example of a matrix with this property.
(8) 14. Find the eigenvalues and eigenvectors of the matrix
\begin{displaymath}A = \left[ \begin{array}{rrr}
-1 & 0 & 0\\
0 & 0 & 2\\
0 & 2 & 3
\end{array} \right] . \end{displaymath}

(80)