Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= -4 \, x + y + 15\]

\[y'= -x - 5 \, y + 9\]

The isoclines cross at \(( 4 , 1 )\), with \(x'=0\) having a positive slope and \(y'=0\) having a negative slope.

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

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= 4 \, x - 5 \, y - 25\]

\[y'= -2 \, x - 2 \, y + 8\]

The isoclines cross at \(( 5 , -1 )\), with \(x'=0\) having a positive slope and \(y'=0\) having a negative slope.

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

Draw a phase plane for the following system of ODEs. Include the isoclines \(x'=0,y'=0\) and mark each of the resulting regions with an arrow system describing the directions of trajectories in that region.

\[x'= 3 \, x - 4 \, y + 19\]

\[y'= 4 \, x + 4 \, y - 12\]

The isoclines cross at \(( -1 , 4 )\), 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.

\[x'= -x + 3 \, y + 7\]

\[y'= -3 \, x - 5 \, y - 7\]

The isoclines cross at \(( 1 , -2 )\), with \(x'=0\) having a positive slope and \(y'=0\) having a negative slope.

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

\[x'= -2 \, x + 3 \, y - 10\]

\[y'= -2 \, x - 3 \, y - 10\]

The isoclines cross at \(( -5 , 0 )\), with \(x'=0\) having a positive slope and \(y'=0\) having a negative slope.

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

\[x'= 3 \, x - 4 \, y + 4\]

\[y'= -2 \, x - 4 \, y + 24\]

The isoclines cross at \(( 4 , 4 )\), with \(x'=0\) having a positive slope and \(y'=0\) having a negative slope.

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

\[x'= 5 \, x - 4 \, y - 8\]

\[y'= -3 \, x - 4 \, y + 24\]

The isoclines cross at \(( 4 , 3 )\), with \(x'=0\) having a positive slope and \(y'=0\) having a negative slope.

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

\[x'= -5 \, x - 3 \, y + 35\]

\[y'= x - 5 \, y + 21\]

The isoclines cross at \(( 4 , 5 )\), with \(x'=0\) having a negative slope and \(y'=0\) having a positive slope.

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

\[x'= 2 \, x - 2 \, y - 6\]

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

The isoclines cross at \(( 1 , -2 )\), with \(x'=0\) having a positive slope and \(y'=0\) having a negative slope.

\[x'= -x - 5 \, y\]

\[y'= -x + 2 \, y + 7\]

The isoclines cross at \(( 5 , -1 )\), with \(x'=0\) having a negative slope and \(y'=0\) having a positive slope.

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

\[x'= -x + 2 \, y\]

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

The isoclines cross at \(( 0 , 0 )\), with \(x'=0\) having a positive slope and \(y'=0\) having a negative slope.

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

\[y'= 5 \, x - 2 \, y + 2\]

The isoclines cross at \(( 0 , 1 )\), with \(x'=0\) having a negative slope and \(y'=0\) having a positive slope.

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

\[x'= -4 \, x - y - 8\]

\[y'= -2 \, x + y - 10\]

The isoclines cross at \(( -3 , 4 )\), with \(x'=0\) having a negative slope and \(y'=0\) having a positive slope.

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

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

\[y'= -3 \, x - 2 \, y - 23\]

The isoclines cross at \(( -5 , -4 )\), with \(x'=0\) having a positive slope and \(y'=0\) having a negative slope.

\[x'= -4 \, x - 4 \, y + 16\]

\[y'= -x + 4 \, y - 6\]

The isoclines cross at \(( 2 , 2 )\), with \(x'=0\) having a negative slope and \(y'=0\) having a positive slope.

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

\[x'= -4 \, x - y - 3\]

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

The isoclines cross at \(( -1 , 1 )\), with \(x'=0\) having a negative slope and \(y'=0\) having a positive slope.

\[x'= 4 \, x - 2 \, y + 2\]

\[y'= -5 \, x - 3 \, y - 8\]

The isoclines cross at \(( -1 , -1 )\), with \(x'=0\) having a positive slope and \(y'=0\) having a negative slope.

\[x'= 3 \, x + 4 \, y - 11\]

\[y'= 2 \, x - 5 \, y - 15\]

The isoclines cross at \(( 5 , -1 )\), with \(x'=0\) having a negative slope and \(y'=0\) having a positive slope.

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

\[x'= 2 \, x + 4 \, y - 18\]

\[y'= 3 \, x - 2 \, y + 13\]

The isoclines cross at \(( -1 , 5 )\), with \(x'=0\) having a negative slope and \(y'=0\) having a positive slope.

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

\[x'= 4 \, x - 5 \, y + 14\]

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

The isoclines cross at \(( -1 , 2 )\), with \(x'=0\) having a positive slope and \(y'=0\) having a negative slope.

\[x'= -2 \, x + 4 \, y + 14\]

\[y'= -4 \, x - 5 \, y - 37\]

The isoclines cross at \(( -3 , -5 )\), with \(x'=0\) having a positive slope and \(y'=0\) having a negative slope.

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

\[y'= 5 \, x - 3 \, y + 15\]

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

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

\[x'= -2 \, x - 5 \, y - 28\]

\[y'= -x + 3 \, y + 8\]

The isoclines cross at \(( -4 , -4 )\), with \(x'=0\) having a negative slope and \(y'=0\) having a positive slope.

\[x'= -2 \, x + 2 \, y - 10\]

\[y'= -3 \, x - 5 \, y + 17\]

The isoclines cross at \(( -1 , 4 )\), with \(x'=0\) having a positive slope and \(y'=0\) having a negative slope.

\[x'= 5 \, x + 2 \, y - 6\]

\[y'= 4 \, x - 2 \, y + 6\]

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

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

\[x'= 5 \, x - 5 \, y - 5\]

\[y'= -4 \, x - 3 \, y + 25\]

The isoclines cross at \(( 4 , 3 )\), with \(x'=0\) having a positive slope and \(y'=0\) having a negative slope.

\[x'= 2 \, x - y + 11\]

\[y'= 4 \, x + y + 13\]

The isoclines cross at \(( -4 , 3 )\), 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.

\[x'= -4 \, x - 4 \, y + 16\]

\[y'= 2 \, x - 2 \, y + 12\]

The isoclines cross at \(( -1 , 5 )\), with \(x'=0\) having a negative slope and \(y'=0\) having a positive slope.

\[x'= 3 \, x - 2 \, y - 7\]

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

The isoclines cross at \(( 3 , 1 )\), with \(x'=0\) having a positive slope and \(y'=0\) having a negative slope.

\[x'= -4 \, x - 4 \, y + 4\]

\[y'= -x + 4 \, y - 14\]

The isoclines cross at \(( -2 , 3 )\), with \(x'=0\) having a negative slope and \(y'=0\) having a positive slope.

\[x'= 3 \, x + 3 \, y + 9\]

\[y'= x - 5 \, y + 15\]

The isoclines cross at \(( -5 , 2 )\), with \(x'=0\) having a negative slope and \(y'=0\) having a positive slope.

\[x'= -5 \, x - 4 \, y + 4\]

\[y'= 5 \, x - 2 \, y + 2\]

The isoclines cross at \(( 0 , 1 )\), with \(x'=0\) having a negative slope and \(y'=0\) having a positive slope.

\[x'= 4 \, x - y + 19\]

\[y'= -4 \, x - 4 \, y - 24\]

The isoclines cross at \(( -5 , -1 )\), with \(x'=0\) having a positive slope and \(y'=0\) having a negative slope.

\[x'= -2 \, x - 5 \, y + 22\]

\[y'= 5 \, x - 3 \, y + 7\]

The isoclines cross at \(( 1 , 4 )\), with \(x'=0\) having a negative slope and \(y'=0\) having a positive slope.

\[x'= 5 \, x - 4 \, y + 31\]

\[y'= -5 \, x - 4 \, y + 1\]

The isoclines cross at \(( -3 , 4 )\), with \(x'=0\) having a positive slope and \(y'=0\) having a negative slope.

\[x'= -x - 3 \, y + 1\]

\[y'= 2 \, x - 2 \, y - 10\]

The isoclines cross at \(( 4 , -1 )\), with \(x'=0\) having a negative slope and \(y'=0\) having a positive slope.

\[x'= -3 \, x - y + 11\]

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

The isoclines cross at \(( 3 , 2 )\), with \(x'=0\) having a negative slope and \(y'=0\) having a positive slope.

\[x'= -5 \, x + y - 12\]

\[y'= -4 \, x - y - 15\]

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

\[x'= 2 \, x + 5 \, y + 9\]

\[y'= 2 \, x - y - 9\]

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

\[x'= -x - 3 \, y - 7\]

\[y'= -4 \, x + y - 2\]

The isoclines cross at \(( -1 , -2 )\), with \(x'=0\) having a negative slope and \(y'=0\) having a positive slope.

\[x'= 2 \, x - 3 \, y + 19\]

\[y'= -5 \, x - 4 \, y + 10\]

The isoclines cross at \(( -2 , 5 )\), with \(x'=0\) having a positive slope and \(y'=0\) having a negative slope.

\[x'= 2 \, x + y - 2\]

\[y'= 4 \, x - 2 \, y + 12\]

The isoclines cross at \(( -1 , 4 )\), with \(x'=0\) having a negative slope and \(y'=0\) having a positive slope.

\[x'= 5 \, x - 5 \, y + 25\]

\[y'= -4 \, x - 3 \, y - 6\]

The isoclines cross at \(( -3 , 2 )\), with \(x'=0\) having a positive slope and \(y'=0\) having a negative slope.

\[x'= -2 \, x + 4 \, y + 10\]

\[y'= -x - 3 \, y - 20\]

The isoclines cross at \(( -5 , -5 )\), with \(x'=0\) having a positive slope and \(y'=0\) having a negative slope.

\[x'= -4 \, x - y + 17\]

\[y'= 2 \, x - 2 \, y - 6\]

The isoclines cross at \(( 4 , 1 )\), with \(x'=0\) having a negative slope and \(y'=0\) having a positive slope.

\[x'= x - y + 5\]

\[y'= -2 \, x - 5 \, y + 11\]

The isoclines cross at \(( -2 , 3 )\), with \(x'=0\) having a positive slope and \(y'=0\) having a negative slope.

\[x'= 2 \, x - y - 10\]

\[y'= -2 \, x - y + 6\]

The isoclines cross at \(( 4 , -2 )\), with \(x'=0\) having a positive slope and \(y'=0\) having a negative slope.

\[x'= 4 \, x + y + 16\]

\[y'= 5 \, x - 4 \, y + 20\]

The isoclines cross at \(( -4 , 0 )\), with \(x'=0\) having a negative slope and \(y'=0\) having a positive slope.

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

\[y'= -5 \, x + 4 \, y + 12\]

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

\[x'= 4 \, x - 3 \, y - 19\]

\[y'= -2 \, x - 2 \, y - 8\]

The isoclines cross at \(( 1 , -5 )\), with \(x'=0\) having a positive slope and \(y'=0\) having a negative slope.