MAT 343 EXAM 3
(due Tue. next week in class)

NAME:______________________

ASU ID:_____________________

1. Find the general solutions for the linear systems of the form

$$\dot x = M x$$

where $M$ takes the matrix values $A,\;\;B,\;\;C$ defined as follows.

$$A=\left( \begin{array}{ccc} 4 & 1 & 1 \\ 2 & 4 & 0 \\ ... ... 0 & 0 & 0 & 1\\ -27 & 0 & 0 & -28 & 0 & 0 \end{array}\right)$$

2. Classify the following second order curves and sketch them in the original coordinate system $(x,y).$

$$x^2 + 6 xy + 9 y^2 + x + y + 1 =0$$

$$\frac{x^2}{2} +\sqrt{3}xy - \frac{y^2}{2} + x = 0$$

$$x^2 +4xy + y^2 + x +y = 0$$

3. Find the canonical form for the following second order hypersurface.

$$x_1^2 + x_2^2 + x_3^2 + x_4^2+ 2\cdot (x_1\cdot x_2 + x_1\cd... ...ot x_4 + x_2\cdot x_3 + x_2\cdot x_4 + x_3 \cdot x_4) + x_1 =0$$