DEPARTMENT OF MATHEMATICS & STATISTICS

MATH 3503

FINAL EXAMINATION
April 1997
TIME: 3 HOURS
 

NO CALCULATORS PERMITTED

MARKS
1. Find the Laplace transforms of the following functions:
(5) (a) ${\displaystyle f(t) = \left\{ \begin{array}{ccc}
t^2 + 1 & , & 0 \leq t \leq 2\\
3 & , & t > 2
\end{array} \right. ; }$
(3) (b) $t^2 \cosh at$.
(6) 2. Find the inverse Laplace transform of
\begin{displaymath}\frac{s^2 - 6}{s^3 + 4s^2 + 3s}\;\;.
\end{displaymath}
(8) 3. Use Laplace transforms to solve the differential equation
\begin{displaymath}y'' + 2y' + y = 3te^{-t},\;\;y(0) = 4,\;\;y'(0) = 2.
\end{displaymath}
(8) 4. Use Laplace transforms to solve the differential equation
\begin{displaymath}y'' + 4y' + 13y = f(t),\;\;y(0) = A,\;\;y'(0) = B.
\end{displaymath}
(10) 5. Use the Method of Frobenius to find a solution to the differential equation
x2y'' + x(1-x)y' - (1+3x)y = 0
about x = 0. State the form of a second linearly independent solution.
(4) 6. Use the identities
\begin{displaymath}\frac{d}{dx} [x^pJ_p(x)] = x^{p}J_{p-1}(x)
\end{displaymath}
and
\begin{displaymath}\frac{d}{dx} [x^{-p}J_p(x)] = -x^{-p}J_{p+1}(x)
\end{displaymath}
to find the recurrence formula for Bessel functions.
(7) 7. Use matrix methods to solve the initial value problem
\begin{displaymath}\frac{d \vec{x}}{dt} = \left[ \begin{array}{rr}
0 & 1\\
-4 &...
...}(0) = \left[ \begin{array}{r}
2\\ -1
\end{array} \right]\;\;. \end{displaymath}
(8) 8. Use matrix methods to find the general solution to
\begin{displaymath}\frac{d \vec{x}}{dt} = \left[ \begin{array}{rr}
2 & 1\\
-4 &...
...t[ \begin{array}{c}
3e^{2t}\\ te^{2t}
\end{array} \right]\;\;. \end{displaymath}
(6) 9. Find the Fourier series for the function
\begin{displaymath}f(x) = \left\{ \begin{array}{ccr}
2 & , & -2 < x \leq 0\\
x & , & 0 < x < 2
\end{array} \right.\;\;. \end{displaymath}
Sketch the graph of the function to which the series converges over the interval $-6 \leq x \leq 6$.
(5) 10. Find the Fourier sine series for the function
\begin{displaymath}f(x) = x^2,\;\;\;0 < x < \pi.
\end{displaymath}
Sketch the graph of the function to which the series converges over the interval $-3 \pi \leq x \leq 3 \pi$.
(10) 11. Use the method of separation of variables to solve the partial differential equation
\begin{displaymath}\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}
= 0
\end{displaymath}
subject to
\begin{displaymath}\left\{ \begin{array}{rcl}
u(0,y) & = & 0\\
u(\ell, y) & = &...
...
u(x,b) & = & 0\\
u(x,0) & = & f(x)
\end{array} \right.\;\;. \end{displaymath}

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