Exam 17: Mathematical Problems and Solutions

Let $A = \left( \begin{array} { c c } - 4 & - 9 \\4 & 8\end{array} \right)$ , and consider the system $X ^ { \prime } = A X$ . The critical point $( 0,0 )$ of the system is a

Let $A = \left( \begin{array} { c c } - 1 & - 4 \\1 & - 1\end{array} \right)$ , and consider the system $X ^ { \prime } = A X$ . The critical point $( 0,0 )$ of the system is a spiral point. The origin is

The solution of $y ^ { \prime \prime } + y = \tan x$ is

Consider the heat problem $k \frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } = \frac { \partial u } { \partial t } , 0 < x < \infty , t > 0 , u ( x , 0 ) = 0 , u ( 0 , t ) = u _ { 0 }$ . Apply a Fourier sine transform. The resulting problem for $U ( \alpha , t ) = \mathcal { F } _ { s } \{ u ( x , t ) \}$ is

The solutions for $\lambda , R \text { and } \Theta$ from the previous problem are

In the previous problem, the exact solution of the initial value problem is

The solution of $y ^ { \prime \prime } + 2 y ^ { \prime } = x + e ^ { x }$ is

In the previous two problems, the solution for u along the line $t = 0.5$ at the mesh points is Select all that apply.

The solution of $y ^ { \prime } + y = x$ is

Using the classical Runge-Kutta method of order 4 with a step size of $h = 0.1$ , the solution of $y ^ { \prime } = 1 + y ^ { 2 } , y ( 0 ) \text { at } x = 0.1$ is

In the previous two problems, the error in the improved Euler method at $x = 0.1$ is

The solution of $y ^ { \prime \prime } - 6 y ^ { \prime } + 8 y = 0$ is

Using the improved Euler method with a step size of $h = 0.1$ , the solution of $y ^ { \prime } = 1 + y ^ { 2 } , y ( 0 ) = 0 \text { at } x = 0.1$ is

In the previous problem, the solution for the temperature is

The eigenvalue-eigenvector pairs for the matrix $A = \left( \begin{array} { c c c } 4 & 0 & 0 \\0 & 3 & 1 \\0 & - 1 & 1\end{array} \right) \mathrm { X }$ are Select all that apply.

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

A 4-pound weight is hung on a spring and stretches it 1 foot. The mass spring system is then put into motion in a medium offering a damping force numerically equal to the velocity. If the mass is pulled down 6 inches from equilibrium and released, the initial value problem describing the position, $x ( t )$ , of the mass at time t is

The solution of $y ^ { \prime \prime } - 4 y ^ { \prime } + 20 y = 0$ is

The solution of $x ^ { 2 } y ^ { \prime \prime } - x y ^ { \prime } = 0$ is

The solutions of the eigenvalue problem and the other problem from the previous problem are