Exam 17: Mathematical Problems and Solutions

Consider the non-linear system $x ^ { \prime } = 1 - 2 x y , y ^ { \prime } = 2 x y - y$ . The linearized system about the one critical point, $( 1 / 2,1 ) \text {, is } X ^ { t } = A X \text {, where } A =$

The solution of $X ^ { \prime } = \left( \begin{array} { c c c } 4 & 0 & 0 \\0 & 3 & 1 \\0 & - 1 & 1\end{array} \right) X$ are

The solutions of a regular Sturm-Liouville problem $\left( \left( r y ^ { \prime } \right) ^ { \prime } + ( \lambda p + q ) y = 0 , y ( a ) = 0 , y ( b ) = 0 \right)$ have which of the following properties?

Consider the heat problem $c \frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } = \frac { \partial u } { \partial t } , u ( 0 , t ) = 0 , u ( 1 , t ) = 3 , u ( x , 0 ) = 3 x ^ { 2 }$ . Replace $\frac { \partial ^ { 2 } u } { \partial x ^ { 2 } }$ with a central difference approximation with $h = 1 / 3$ and $\frac { \partial u } { \partial t }$ with a forward difference approximation with $k = 1 / 2$ . The resulting equation is

Using power series methods, the solution of $2 x y ^ { \prime \prime } + y ^ { \prime } + 2 y = 0$ is

Using Laplace transform methods, the solution of $y ^ { \prime \prime } + y = \delta ( t - \pi / 2 ) , y ( 0 ) = 1$ , $y ^ { \prime } ( 0 ) = 0$ is

The solution of $y ^ { \prime \prime } + 3 y ^ { \prime } - 4 y = \cos x$ is

Using the convolution theorem, we find that $\mathcal { L } ^ { - 1 } \left\{ 1 / \left( ( s + 1 ) \left( s ^ { 2 } + 1 \right) \right) \right\} =$