Calculus HW
mirananwar
Hw19.pdf
Miran Anwar, mth 235 ss21 1: Hw19-5.4-SDE-2x2NS-PP-NP. Due: 04/26/2021 at 11:00pm EDT. See in LN, § 5.4, See Subsection 5.4.3. 1. (10 points) Part 1: Critical Points Consider the two-dimensional autonomous system
x′ =− 1 2
x + xy
y′ = 3 2
y− 1 2
xy
(a) The critical points of the system above have the form
x0 = [
0 0
] , x1 =
[ x1 y1
] .
Find these components.
x1 =
y1 =
Part 2: The Jacobian Matrix Part 3: The Jacobian Matrix at X0 Part 4: The Jacobian Matrix at X1
See in LN, § 5.4, See Subsection 5.4.3. 2. (10 points) Part 1: Critical Points Consider the two-dimensional autonomous system
x′ =− 1 4
x + 1 2
xy
y′ = y− 1 2
xy
(a) The critical points of the system above have the form
x0 = [
0 0
] , x1 =
[ x1 y1
] .
Find these components.
x1 =
y1 =
Part 2: The Jacobian Matrix Part 3: The Jacobian Matrix at X0
1
Part 4: The Jacobian Matrix at X1
See in LN, § 5.4, See Subsection 5.4.4. 3. (10 points) Part 1: Critical Points Consider the two-dimensional autonomous system
x′ =− 1 4
x + 1 2
xy
y′ = y− 1 2
y2 − 1 2
xy
(a) The critical points of the system above have the form
x0 = [
0 0
] , x1 =
[ 0 y1
] , x2 =
[ x2 y2
] .
Find these components.
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
See in LN, § 5.4, See Subsection 5.4.4. 4. (10 points) Part 1: Critical Points Consider the two-dimensional autonomous system
x′ =−x + xy
y′ = 9 8
y− y2 − 1 2
xy
(a) The critical points of the system above have the form
x0 = [
0 0
] , x1 =
[ 0 y1
] , x2 =
[ x2 y2
] .
Find these components.
y1 =
x2 =
y2 = 2
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, § 5.4, See Example 5.4.5. 5. (10 points)
Nonlinear Pendulum: No Friction Case The equation of a pendulum having an attached mass m > 0, a massless rod of length l > 0 is
mlθ′′ =−mg sin(θ),
where θ(t) is the angular position of the pendulum as function of time, measured from the vertical down- wards position, positive counter-clockwise, and g is the acceleration of gravity. We consider the particular case
g l = 1 and m = 1.
Part 1: First Order Reduction (a) Find the first order reduction of the equation above where u = θ and v = θ′.
u′ =
v′ =
Part 2: Critical Points Part 3: The Derivative Matrix Part 4: The Derivative Matrix at Even Critical Points Part 5: The Derivative Matrix at Odd Critical Points
See in LN, § 5.4, See Example 5.4.5. 6. (10 points)
Nonlinear Pendulum: Small Friction Case The equation of a pendulum having an attached mass m > 0, a massless rod of length l > 0, and swinging in a medium with damping constant d > 0 is
mlθ′′ =−mg sin(θ)−dlθ′,
where θ(t) is the angular position of the pendulum as function of time, measured from the vertical down- wards position, positive counter-clockwise, and g is the acceleration of gravity. We consider the following particular case:
• g l = 1 and m = 1.
• Small friction, 0 < d < 2. 3
Part 1: First Order Reduction (a) Find the first order reduction of the equation above where u = θ and v = θ′.
u′ =
v′ =
Part 2: Critical Points Part 3: The Derivative Matrix Part 4: The Derivative Matrix at Even Critical Points Part 5: The Derivative Matrix at Odd Critical Points See in LN, § 5.4, See Example 5.4.5. 7. (10 points)
Nonlinear Pendulum: Large Friction Case The equation of a pendulum having an attached mass m > 0, a massless rod of length l > 0, and swinging in a medium with damping constant d > 0 is
mlθ′′ =−mg sin(θ)−dlθ′,
where θ(t) is the angular position of the pendulum as function of time, measured from the vertical down- wards position, positive counter-clockwise, and g is the acceleration of gravity. We consider the following particular case:
• g l = 1 and m = 1.
• Large friction, d > 2.
Part 1: First Order Reduction (a) Find the first order reduction of the equation above where u = θ and v = θ′.
u′ =
v′ =
Part 2: Critical Points Part 3: The Derivative Matrix Part 4: The Derivative Matrix at Even Critical Points Part 5: The Derivative Matrix at Odd Critical Points
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