DEPARTMENT OF MATHEMATICS & STATISTICS

MATH 2503

FINAL EXAMINATION
April 1997
TIME: 3 HOURS

NO CALCULATORS

MARKS
(12) 1. Solve the following differential equations :
(a) ${\displaystyle (1 + y^2)\; \frac{dy}{dx} = ye^{2x}}$
(b) ${\displaystyle x^2\; \frac{dy}{dx} + 3xy = 1}$
(c) ${\displaystyle \frac{dy}{dx} = \frac{3y^2 - x^2}{2xy}}$
(8)2. Find the general solution for the following differential equations:
(a) y'' + 2y' + 2y = 0
(b) y'' + y' = 2x + 1
(6)3. Solve the following initial value problem:
\begin{displaymath}y'' - y' - 2y = -4e^x,\;\;\;y(0) = 3,\;\;\;y'(0) = -2.
\end{displaymath}
(16)4. (a) Solve, using the method of variation of parameters $y'' + y = \tan x$.
(b) Use power series to solve y'' + xy' + y = 0.
(12)5. Test the following series for convergence or divergence:
(a) ${\displaystyle \sum_{n=0}^{\infty}\;\frac{n+1}{3n^3 + 1}}$ (b) ${\displaystyle \sum_{n=1}^{\infty}\;2^{-1/n}}$ (c) ${\displaystyle \sum _{n=2}^{\infty}\;\frac{1}{n \ln
n}}$
(15)6. (a) Test whether, the following series is absolutely, conditionally convergent or divergent:
\begin{displaymath}\sum_{n=1}^{\infty}\;\frac{(-1)^{n-1}}{\sqrt{n^2 + 1}}\;\;.
\end{displaymath}
(b) Find the interval of convergence of the series
\begin{displaymath}\sum_{n=0}^{\infty}\;\frac{(-1)^n(x-3)^n}{2^{2n}}\;\;.
\end{displaymath}
(c) Find the Maclauren series for ${\displaystyle
\frac{1}{(1-x)^2}}$.
(5)7. Use Gauss-Jordan elimination method to find all the solutions, if any, of the following system
x + 3y + z = 2
x - 2y - 2z = 3
2x + y - z = 5
3x - y - 3z = 8.
(8)8. (a) Find the inverse of the matrix
\begin{displaymath}A = \left[ \begin{array}{ccc}
1 & 2 & 3\\
1 & 1 & 2\\
0 & 1 & 2
\end{array} \right]. \end{displaymath}
(b) Use (a) to solve
x + 2y + 3z = 0  
x + y + 2z = 1  
    y + 2z = 2.  
(8)9. (a) Evaluate the following determinant:
\begin{displaymath}\left\vert \begin{array}{rrrr}
1 & 2 & 0 & 0\\
3 & -2 & 1 & 0\\
0 & 0 & 2 & -5\\
0 & 0 & 1 & 1
\end{array} \right\vert. \end{displaymath}
(b) Use Cramer's rule to find the value of z in the following system:
x - 2y + z = 0
    y - 2z = 0
x + 3y - z = 1.
(10)10. Let ${\displaystyle A = \left[ \begin{array}{rrr}
0 & 0 & -1\\
4 & 0 & 0\\
3 & -3 & 4
\end{array} \right]}$.
(a) Find the characteristic equation of A.
(b) Show 4 is an eigenvalue of A.
(c) Find an eigenvector of A corresponding to the eigenvalue 4.

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