Calculus HW
Miran Anwar, mth 235 ss21 1: Hw18-5.3-5.4-SDE-2x2NS-CS. Due: 04/26/2021 at 11:00pm EDT.
Similar problem in LN, § 5.3, Example 5.3.6, 5.3.7. 1. (10 points)
Find the critical points (also called equilibrium solutions) of the predator-prey system
x′ =−3 x + 2 x y y′ = 7 y−8 x y
Critical Points:
Note: A point is an ordered pair (x,y), and your answer must be a comma separated list of points.
Similar problem in LN, § 5.3, Example 5.3.6, 5.3.7. 2. (10 points)
Find the critical points (also called equilibrium solutions) of the competing species system
x′ = 3 x−x2 −2 x y y′ = 2 y−y2 −x y
Equilibrium Points:
Note: A point is an ordered pair (x,y), and your answer must be a comma separated list of points.
See in LN, § 5.3, See Examples 5.3.8-5.3.10. 3. (10 points) Part 1: Critical Points Consider the two-dimensional autonomous system
x′ =−9 x + x3
y′ =−2 y
(a) The critical points of the system above have the form
x0 = [
0 0
] , x1 =
[ x1 0
] , x2 =
[ x2 0
] .
where x1 > 0 > x2. Find these components.
x1 = x2 =
Part 2: The Jacobian Matrix 1
Part 3: The Jacobian Matrix at X0 Part 4: The Jacobian Matrix at X1 Part 5: The Jacobian Matrix at X2 See in LN, § 5.3, See Examples 5.3.8-5.3.10. 4. (10 points) Part 1: Critical Points Consider the two-dimensional autonomous system
x′ =−16 x + x3
y′ = 3 y
(a) The critical points of the system above have the form
x0 = [
0 0
] , x1 =
[ x1 0
] , x2 =
[ x2 0
] .
where x1 > 0 > x2. Find these components.
x1 = x2 =
Part 2: The Jacobian Matrix Part 3: The Jacobian Matrix at X0 Part 4: The Jacobian Matrix at X1 Part 5: The Jacobian Matrix at X2 See in LN, § 5.3, See Examples 5.3.8-5.3.10. 5. (10 points) Part 1: Critical Points Consider the two-dimensional autonomous system
x′ = 4 y−y3
y′ =−9 x−y2
(a) The critical points of the system above have the form
x0 = [
0 0
] , x1 =
[ x1 y1
] , x2 =
[ x2 y2
] .
where y1 > 0 > y2. Find these components.
x1 = y1 =
x2 = y2 =
Part 2: The Jacobian Matrix Part 3: The Jacobian Matrix at X0 Part 4: The Jacobian Matrix at X1 Part 5: The Jacobian Matrix at X2
2
See in LN, § 5.4, See Examples 5.4.1, 5.4.2. 6. (10 points) Part 1: Critical Points Consider the two-dimensional autonomous system
x′ = x (1− x− y)
y′ = y (1
2 −
1 4
y− 3 4
x )
(a) The critical points of the system above have the form
x0 = [
0 0
] , x1 =
[ 0 y1
] , x2 =
[ x2 0
] , x3 =
[ x3 y3
] .
where y1 > 0 > y2. Find these components.
y1 = x2 =
x3 = y3 =
Part 2: The Jacobian Matrix Part 3: The Jacobian Matrix at X0 Part 4: The Jacobian Matrix at X1 Part 5: The Jacobian Matrix at X2 Part 6: The Jacobian Matrix at X3 See in LN, § 5.4, See Examples 5.4.1, 5.4.2. 7. (10 points) Part 1: Critical Points Consider the two-dimensional autonomous system
x′ = x (1− x− y)
y′ = y (3
4 − y−
1 2
x )
(a) The critical points of the system above have the form
x0 = [
0 0
] , x1 =
[ 0 y1
] , x2 =
[ x2 0
] , x3 =
[ x3 y3
] .
where y1 > 0 > y2. Find these components.
y1 = x2 =
x3 = y3 =
Part 2: The Jacobian Matrix Part 3: The Jacobian Matrix at X0 Part 4: The Jacobian Matrix at X1
3
Part 5: The Jacobian Matrix at X2 Part 6: The Jacobian Matrix at X3
See in LN, § 5.4, See Examples 5.4.1, 5.4.2. 8. (10 points) Part 1: Critical Points Consider the two-dimensional autonomous system
x′ = x ( 3 2 − x−
1 2
y)
y′ = y (
2− y− 3 4
x )
(a) The critical points of the system above have the form
x0 = [
0 0
] , x1 =
[ 0 y1
] , x2 =
[ x2 0
] , x3 =
[ x3 y3
] .
where y1 > 0 > y2. Find these components.
y1 = x2 =
x3 = y3 =
Part 2: The Jacobian Matrix Part 3: The Jacobian Matrix at X0 Part 4: The Jacobian Matrix at X1 Part 5: The Jacobian Matrix at X2 Part 6: The Jacobian Matrix at X3
See in LN, § 5.4, See Examples 5.4.1, 5.4.2. 9. (10 points) Part 1: Critical Points Consider the two-dimensional autonomous system
x′ = x ( 3 2 − x−
1 2
y)
y′ = y (
2− 1 2
y− 3 2
x )
(a) The critical points of the system above have the form
x0 = [
0 0
] , x1 =
[ 0 y1
] , x2 =
[ x2 0
] , x3 =
[ x3 y3
] .
where y1 > 0 > y2. Find these components.
y1 = x2 =
x3 = y3 = 4
Part 2: The Jacobian Matrix Part 3: The Jacobian Matrix at X0 Part 4: The Jacobian Matrix at X1 Part 5: The Jacobian Matrix at X2 Part 6: The Jacobian Matrix at X3
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