Draw a phase plane for the following system of ODEs.

\[x'= 3 \, x - 2 \, y - 15 \]

\[y'= 5 \, x + 3 \, y - 6 \]

Be sure to include and label the following features in your sketch:

The isoclines \(x'=0\) and \(y'=0\).
Arrow systems describing trajectories of solutions.
The equillibrium of the system.

Answer:

The isoclines cross at the equillibrium \( \left(3, -3\right) \), with \(x'=0\) having a positive slope and \(y'=0\) having a negative slope.

The right region has a northeast arrow system, the top region has a northwest arrow system, the left region has a southwest arrow system, and the bottom region has a southeast arrow system.