Exam 16: Mathematics Problems: Differential Equations and Linear Algebra

The solution of $x ^ { 2 } y ^ { \prime \prime } + 3 x y ^ { \prime } - 3 y = 0$ is

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

The differential equation $y ^ { \prime } + y = x ^ { 2 }$ is Select all that apply.

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

The solution of $\mathbf { X } ^ { \prime } = \left( \begin{array} { l l } 1 & - 2 \\1 & - 1\end{array} \right) \mathbf { X }$ is

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

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

In the previous two problems, the solution for the temperature is

The solution of the previous problem is

The auxiliary equation of $y ^ { \prime \prime } - 5 y ^ { \prime } + 6 y = 0$ is

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

The solution of the system in the previous problem is

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

The solution of $x ^ { 2 } y d x + \left( x ^ { 3 } / 3 + y \right) d y = 0$ is

The eigenvalues of the matrix $A = \left( \begin{array} { l l } 1 & - 2 \\1 & - 1\end{array} \right)$ are

A uniform beam of length 10 has a concentrated load $w _ { 0 }$ at $x = 5$ . It is embedded at both ends. The boundary value problem for the deflections, $y ( x )$ , for this system is

Let $A = \left( \begin{array} { c c } 2 & 0 \\0 & - 3\end{array} \right)$ . Then $e ^ { A t } =$

The differential equation $y ^ { \prime } = x ^ { 2 } y ^ { 2 }$ is Select all that apply.

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

The solution of the eigenvalue problem $y ^ { \prime \prime } + \lambda y = 0 , y ^ { \prime } ( 0 ) = 0 , y ( 1 ) = 0$ is