MATH528Oct 17, 2018

Instructions: Exam is closed book. Present your answers legibly, show and briefly comment steps in your solution.

1. Find the general solution of the following ODEs:

   1. $y^{\text{'}}+2.5y=1.6x$

   2. $y^{\text{'}}-0.4y=29\sin x$

   3. $25y{y}^{\text{'}}-4x=0$

   Classify each equation. ( $3\times 2$ points = 6 points)

2. Solve the IVP $y^{\text{'}\text{'}\text{'}}-y^{\text{'}\text{'}}-y^{\text{'}}+y=0$, $y\left(0\right)=0,y^{\text{'}}\left(0\right)=1,y^{\text{'}\text{'}}\left(0\right)=0.$ (3 points)

3. Find the general solution and determine the type of critical points of the system

   ${\displaystyle \begin{array}{l}{y}_1^{\text{'}}=3y_1+4y_2\\ y_2^{\text{'}}=3y_1+2y_2\end{array}.}$

   (3 points)