DEPARTMENT OF MATHEMATICS & STATISTICS

MATH 3503

FINAL EXAMINATION
April 1998
TIME: 3 HOURS
 

NO CALCULATORS PERMITTED

MARKS
(3) 1. Use the definition of the Laplace transform to show that
\begin{displaymath}{\cal L}\{tf(t)\} = -\;\frac{d}{ds} {\cal L}\{f(t)\}.
\end{displaymath}
(5) 2. Find the Laplace transform of
\begin{displaymath}f(t) = \left\{ \begin{array}{ccc}
1 & , & 0 \leq t \leq 2\\
...
...& 2 < t < 4\\
\sin t & , & t \geq 4
\end{array} \right. \;\;. \end{displaymath}
(5) 3. Find the inverse Laplace transform of
\begin{displaymath}F(s) = \left( \frac{s^2 + 5s - 4}{s^3 + 3s^2 + 2s} \right) e^{-3s}.
\end{displaymath}
(8) 4. Use Laplace transforms to solve the differential equation
\begin{displaymath}y'' + 2y' + 2y = e^{-t} \cos t,\;\;y(0) = A,\;\;y'(0) = B.
\end{displaymath}
(8) 5. Use the Method of Frobenius to find a solution to the differential equation
x2y'' + (x2 - 3x)y' + 4y = 0
about x = 0. State the form of a second linearly independent solution.
(4) 6. Use the substitution y = x-1/2u(x) to convert the differential equation
xy'' + 2y' + xy = 0
to a Bessel equation. Use this to solve the differential equation.
(7) 7. Use matrix methods to solve the system
\begin{displaymath}\frac{d \vec{x}}{dt} = \left[ \begin{array}{rrr}
3 & 0 & 1\\
9 & -1 & 2\\
-9 & 4 & -1
\end{array} \right] \vec{x} \end{displaymath}
given that the characteristic equation is $(3-\lambda)(\lambda^2 + 2
\lambda + 2) = 0$.
(7) 8. Use matrix methods to solve the system
\begin{displaymath}\frac{d \vec{x}}{dt} = \left[ \begin{array}{rr}
4 & 2\\
3 & ...
...\;\;\left[ \begin{array}{c}
0\\ e^{-2t}
\end{array} \right]\;. \end{displaymath}
(5) 9. Find the Fourier series for the function
\begin{displaymath}f(x) = \vert x\vert,\;\;-2 < x \leq 2,\;\;f(x+4) = f(x).
\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) = \pi - x,\;\;\;0 < x < \pi.
\end{displaymath}
Sketch the graph of the function to which the series converges on $-3
\pi \leq x \leq 3 \pi$.
(8) 11. Use the method of separation of variables to solve the partial differential equation
\begin{displaymath}\frac{\partial^2 u}{\partial x^2} = \frac{1}{k^2}\;\frac{\partial
u}{\partial t} \hspace*{1cm} 0 < x < \ell,\;\;\;t > 0
\end{displaymath}
subject to
\begin{eqnarray*}u(x,0) & = & f(x)\\
u_x(0,t) & = & u_x(\ell,t) = 0.
\end{eqnarray*}

(65)