S3: Phase planes


Example 1

S3: Phase planes (ver. 1)

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.

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

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

Answer.

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.


Example 2

S3: Phase planes (ver. 2)

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.

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

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

Answer.

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.


Example 3

S3: Phase planes (ver. 3)

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.

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

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

Answer.

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.


Example 4

S3: Phase planes (ver. 4)

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.

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

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

Answer.

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.


Example 5

S3: Phase planes (ver. 5)

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.

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

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

Answer.

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.


Example 6

S3: Phase planes (ver. 6)

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.

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

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

Answer.

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.


Example 7

S3: Phase planes (ver. 7)

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.

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

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

Answer.

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.


Example 8

S3: Phase planes (ver. 8)

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.

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

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

Answer.

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.


Example 9

S3: Phase planes (ver. 9)

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.

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

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

Answer.

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 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.


Example 10

S3: Phase planes (ver. 10)

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.

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

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

Answer.

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.


Example 11

S3: Phase planes (ver. 11)

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.

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

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

Answer.

The isoclines cross at \(( 0 , 0 )\), 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.


Example 12

S3: Phase planes (ver. 12)

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.

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

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

Answer.

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.


Example 13

S3: Phase planes (ver. 13)

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.

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

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

Answer.

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.


Example 14

S3: Phase planes (ver. 14)

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.

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

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

Answer.

The isoclines cross at \(( -5 , -4 )\), 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.


Example 15

S3: Phase planes (ver. 15)

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.

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

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

Answer.

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.


Example 16

S3: Phase planes (ver. 16)

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.

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

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

Answer.

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


Example 17

S3: Phase planes (ver. 17)

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.

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

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

Answer.

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


Example 18

S3: Phase planes (ver. 18)

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.

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

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

Answer.

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.


Example 19

S3: Phase planes (ver. 19)

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.

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

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

Answer.

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.


Example 20

S3: Phase planes (ver. 20)

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.

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

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

Answer.

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 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.


Example 21

S3: Phase planes (ver. 21)

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.

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

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

Answer.

The isoclines cross at \(( -3 , -5 )\), 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.


Example 22

S3: Phase planes (ver. 22)

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.

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

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

Answer.

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.


Example 23

S3: Phase planes (ver. 23)

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.

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

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

Answer.

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


Example 24

S3: Phase planes (ver. 24)

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.

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

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

Answer.

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 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.


Example 25

S3: Phase planes (ver. 25)

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.

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

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

Answer.

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.


Example 26

S3: Phase planes (ver. 26)

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.

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

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

Answer.

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.


Example 27

S3: Phase planes (ver. 27)

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.

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

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

Answer.

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.


Example 28

S3: Phase planes (ver. 28)

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.

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

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

Answer.

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 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.


Example 29

S3: Phase planes (ver. 29)

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.

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

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

Answer.

The isoclines cross at \(( 3 , 1 )\), 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.


Example 30

S3: Phase planes (ver. 30)

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.

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

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

Answer.

The isoclines cross at \(( -2 , 3 )\), 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.


Example 31

S3: Phase planes (ver. 31)

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.

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

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

Answer.

The isoclines cross at \(( -5 , 2 )\), 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.


Example 32

S3: Phase planes (ver. 32)

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.

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

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

Answer.

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.


Example 33

S3: Phase planes (ver. 33)

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.

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

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

Answer.

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.


Example 34

S3: Phase planes (ver. 34)

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.

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

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

Answer.

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


Example 35

S3: Phase planes (ver. 35)

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.

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

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

Answer.

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


Example 36

S3: Phase planes (ver. 36)

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.

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

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

Answer.

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


Example 37

S3: Phase planes (ver. 37)

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.

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

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

Answer.

The isoclines cross at \(( 3 , 2 )\), 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.


Example 38

S3: Phase planes (ver. 38)

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.

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

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

Answer.

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


Example 39

S3: Phase planes (ver. 39)

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.

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

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

Answer.

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


Example 40

S3: Phase planes (ver. 40)

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.

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

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

Answer.

The isoclines cross at \(( -1 , -2 )\), 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.


Example 41

S3: Phase planes (ver. 41)

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.

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

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

Answer.

The isoclines cross at \(( -2 , 5 )\), 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.


Example 42

S3: Phase planes (ver. 42)

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.

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

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

Answer.

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


Example 43

S3: Phase planes (ver. 43)

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.

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

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

Answer.

The isoclines cross at \(( -3 , 2 )\), 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.


Example 44

S3: Phase planes (ver. 44)

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.

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

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

Answer.

The isoclines cross at \(( -5 , -5 )\), 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.


Example 45

S3: Phase planes (ver. 45)

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.

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

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

Answer.

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


Example 46

S3: Phase planes (ver. 46)

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.

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

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

Answer.

The isoclines cross at \(( -2 , 3 )\), 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.


Example 47

S3: Phase planes (ver. 47)

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.

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

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

Answer.

The isoclines cross at \(( 4 , -2 )\), 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.


Example 48

S3: Phase planes (ver. 48)

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.

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

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

Answer.

The isoclines cross at \(( -4 , 0 )\), 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.


Example 49

S3: Phase planes (ver. 49)

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.

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

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

Answer.

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 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.


Example 50

S3: Phase planes (ver. 50)

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.

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

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

Answer.

The isoclines cross at \(( 1 , -5 )\), 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.