Calculus HW

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Miran Anwar, mth 235 ss21 1: Hw17-5.2-SDE-2x2PP. Due: 04/26/2021 at 11:00pm EDT.

See in LN, § 5.2, See Example 5.2.4. 1. (10 points)

Note: You have only 5 attempts to solve this problem.

Consider the system of differential equations x′ = A x, where x = [

x1 x2

] and the 2×2 matrix A has eigenpairs

λ± = α±β i, α = 0.25, β = 2,

v± = a±b i, a = [

2 1

] , b =

[ −1/2

] ,

Draw on paper the solution x(t) = eαt ( a cos(βt)−b sin(βt)

) on the x1x2-plane and then answer the ques-

tions below.

(a) Based on your graph, select the correct solution curve from the interactive graph below.

• Select One • Curve 1 • Curve 2 • Curve 3 • Curve 4 • None

(b) Use the graph below to find lim t→+∞

‖x(t)‖, where ‖x(t)‖ is the length of the solution vector x(t).

• Select One • Zero • Infinity • None

u(t),

where u(t) = x(t) ‖x(t)‖

, is a unit vector in the direction of the solution vector x(t).

[Select One/U1/-U1/U2/-U2/None]

(d) Use the graph below to find lim t→−∞

‖x(t)‖. 1

• Select One • Zero • Infinity • None

(e) Use the graph below to find the lim t→−∞

u(t), where u(t) = x(t) ‖x(t)‖

[Select One/U1/-U1/U2/-U2/None]

(f) Characterize the zero solution, x0 = 0.

• Select One • Source Node • Source Spiral • Sink Node • Sink Spiral • Saddle • Center • None

Comments on the graph below: • The graph is interactive. • You can click on the boxes to turn on and off possible solution curves. • For each possible solution we display: the possible solution curve, the possible solution vector

x(t), and the associated unit vector u(t) = x(t)/‖x(t)‖. • You can move the time slider to see how each possible solution vector x(t) and unit vector u(t)

change in time. • You can turn on or off the ellipse formed by vectors a and b.

See in LN, § 5.2, See Example 5.2.4. 2. (10 points)

Note: You have only 5 attempts to solve this problem.

Consider the system of differential equations x′ = A x, where x = [

x1 x2

] and the 2×2 matrix A has eigenpairs

λ± = α±β i, α = 0.45, β = 2,

v± = a±b i, a = − [

2 1

] , b = −

[ −1/2

] ,

Draw on paper the solution x(t) = eαt ( a sin(βt)+ b cos(βt)

) on the x1x2-plane and then answer the ques-

tions below. 2

(a) Based on your graph, select the correct solution curve from the interactive graph below.

• Select One • Curve 1 • Curve 2 • Curve 3 • Curve 4 • None

(b) Use the graph below to find lim t→+∞

‖x(t)‖, where ‖x(t)‖ is the length of the solution vector x(t).

• Select One • Zero • Infinity • None

u(t),

where u(t) = x(t) ‖x(t)‖

, is a unit vector in the direction of the solution vector x(t).

[Select One/U1/-U1/U2/-U2/None]

(d) Use the graph below to find lim t→−∞

‖x(t)‖.

• Select One • Zero • Infinity • None

(e) Use the graph below to find the lim t→−∞

u(t), where u(t) = x(t) ‖x(t)‖

[Select One/U1/-U1/U2/-U2/None]

(f) Characterize the zero solution, x0 = 0.

• Select One • Source Node • Source Spiral • Sink Node • Sink Spiral • Saddle • Center • None

x(t), and the associated unit vector u(t) = x(t)/‖x(t)‖. • You can move the time slider to see how each possible solution vector x(t) and unit vector u(t)

change in time. • You can turn on or off the ellipse formed by vectors a and b.

See in LN, § 5.2, See Example 5.2.4. 3. (10 points)

Note: You have only 5 attempts to solve this problem.

Consider the system of differential equations x′ = A x, where x = [

x1 x2

] and the 2×2 matrix A has eigenpairs

λ± = α±β i, α = −0.4, β = 2,

v± = a±b i, a = [

2 1

] , b = −

[ −1/2

] ,

Draw on paper the solution x(t) = eαt ( a cos(βt)−b sin(βt)

) on the x1x2-plane and then answer the ques-

tions below.

(a) Based on your graph, select the correct solution curve from the interactive graph below.

• Select One • Curve 1 • Curve 2 • Curve 3 • Curve 4 • None

(b) Use the graph below to find lim t→+∞

‖x(t)‖, where ‖x(t)‖ is the length of the solution vector x(t).

• Select One • Zero • Infinity • None

u(t),

where u(t) = x(t) ‖x(t)‖

, is a unit vector in the direction of the solution vector x(t).

[Select One/U1/-U1/U2/-U2/None]

(d) Use the graph below to find lim t→−∞

‖x(t)‖.

• Select One • Zero • Infinity • None

(e) Use the graph below to find the lim t→−∞

u(t), where u(t) = x(t) ‖x(t)‖

[Select One/U1/-U1/U2/-U2/None]

(f) Characterize the zero solution, x0 = 0.

• Select One • Source Node • Source Spiral • Sink Node • Sink Spiral • Saddle • Center • None

x(t), and the associated unit vector u(t) = x(t)/‖x(t)‖. • You can move the time slider to see how each possible solution vector x(t) and unit vector u(t)

change in time. • You can turn on or off the ellipse formed by vectors a and b.

See in LN, § 5.2, See Example 5.2.4. 4. (10 points)

Note: You have only 5 attempts to solve this problem.

Consider the system of differential equations x′ = A x, where x = [

x1 x2

] and the 2×2 matrix A has eigenpairs

λ± = α±β i, α = −0.3, β = 2, 5

v± = a±b i, a = − [

2 1

] , b =

[ −1/2

] ,

Draw on paper the solution x(t) = eαt ( a sin(βt)+ b cos(βt)

) on the x1x2-plane and then answer the ques-

tions below.

(a) Based on your graph, select the correct solution curve from the interactive graph below.

• Select One • Curve 1 • Curve 2 • Curve 3 • Curve 4 • None

(b) Use the graph below to find lim t→+∞

‖x(t)‖, where ‖x(t)‖ is the length of the solution vector x(t).

• Select One • Zero • Infinity • None

u(t),

where u(t) = x(t) ‖x(t)‖

, is a unit vector in the direction of the solution vector x(t).

[Select One/U1/-U1/U2/-U2/None]

(d) Use the graph below to find lim t→−∞

‖x(t)‖.

• Select One • Zero • Infinity • None

(e) Use the graph below to find the lim t→−∞

u(t), where u(t) = x(t) ‖x(t)‖

[Select One/U1/-U1/U2/-U2/None]

(f) Characterize the zero solution, x0 = 0.

• Select One • Source Node • Source Spiral • Sink Node • Sink Spiral • Saddle • Center • None

x(t), and the associated unit vector u(t) = x(t)/‖x(t)‖. • You can move the time slider to see how each possible solution vector x(t) and unit vector u(t)

change in time. • You can turn on or off the ellipse formed by vectors a and b.

See in LN, § 5.2, See Examples 5.2.1, 5.2.2, 5.2.3. 5. (10 points)

Note: You have only 5 attempts to solve this problem.

Consider the system of differential equations x′ = A x, where x = [

x1 x2

] and the 2×2 matrix A has eigenpairs

λ1 = 1.5, v1 = [

1 1

] , and λ2 = 0.75, v2 =

[ −1 1

] .

Draw on paper the solution x(t) =−v1 eλ1t +v2 eλ2t on the x1x2-plane and then answer the questions below.

(a) Based on your graph, select the correct solution curve from the interactive graph below.

• Select One • Curve 1 • Curve 2 • Curve 3 • Curve 4 • None

(b) Use the graph below to find lim t→+∞

‖x(t)‖, where ‖x(t)‖ is the length of the solution vector x(t). 7

• Select One • Zero • Infinity • None

lim t→+∞

u(t), where u(t) = x(t) ‖x(t)‖

, is a unit vector in the direction of the solution vector x(t).

[Select One/U1/-U1/U2/-U2/None]

(d) Use the graph below to find lim t→−∞

‖x(t)‖.

• Select One • Zero • Infinity • None

(e) Use the graph below to find the lim t→−∞

u(t), where u(t) = x(t) ‖x(t)‖

[Select One/U1/-U1/U2/-U2/None]

(f) Characterize the zero solution, x0 = 0.

• Select One • Source Node • Source Spiral • Sink Node • Sink Spiral • Saddle • Center • None

x(t), and the associated unit vector u(t) = x(t)/‖x(t)‖. • You can move the time slider to see how each possible solution vector x(t) and unit vector u(t)

change in time. • You can move the eigenvectors v1 and v2 by dragging them from the endpoint, and then see how

the curves would change.

See in LN, § 5.2, See Examples 5.2.1, 5.2.2, 5.2.3. 6. (10 points)

Note: You have only 5 attempts to solve this problem.

Consider the system of differential equations x′ = A x, where x = [

x1 x2

] and the 2×2 matrix A has eigenpairs

λ1 = 0.5, v1 = [

1 1

] , and λ2 =−1.75, v2 =

[ −1 1

] .

Draw on paper the solution x(t) = v1 eλ1t + v2 eλ2t on the x1x2-plane and then answer the questions below.

(a) Based on your graph, select the correct solution curve from the interactive graph below.

• Select One • Curve 1 • Curve 2 • Curve 3 • Curve 4 • None

(b) Use the graph below to find lim t→+∞

‖x(t)‖, where ‖x(t)‖ is the length of the solution vector x(t).

• Select One • Zero • Infinity • None

lim t→+∞

u(t), where u(t) = x(t) ‖x(t)‖

, is a unit vector in the direction of the solution vector x(t).

[Select One/U1/-U1/U2/-U2/None]

(d) Use the graph below to find lim t→−∞

‖x(t)‖.

• Select One • Zero • Infinity • None

(e) Use the graph below to find the lim t→−∞

u(t), where u(t) = x(t) ‖x(t)‖

[Select One/U1/-U1/U2/-U2/None]

(f) Characterize the zero solution, x0 = 0.

• Select One • Source Node • Source Spiral • Sink Node • Sink Spiral • Saddle • Center • None

x(t), and the associated unit vector u(t) = x(t)/‖x(t)‖. • You can move the time slider to see how each possible solution vector x(t) and unit vector u(t)

change in time. • You can move the eigenvectors v1 and v2 by dragging them from the endpoint, and then see how

the curves would change.

See in LN, § 5.2, See Examples 5.2.1, 5.2.2, 5.2.3. 7. (10 points)

Note: You have only 5 attempts to solve this problem.

Consider the system of differential equations x′ = A x, where x = [

x1 x2

] and the 2×2 matrix A has eigenpairs

λ1 =−1, v1 = [

1 1

] , and λ2 =−1.5, v2 =

[ −1 1

] .

Draw on paper the solution x(t) =−v1 eλ1t +v2 eλ2t on the x1x2-plane and then answer the questions below.

(a) Based on your graph, select the correct solution curve from the interactive graph below.

• Select One • Curve 1 • Curve 2 • Curve 3 • Curve 4 • None

(b) Use the graph below to find lim t→+∞

‖x(t)‖, where ‖x(t)‖ is the length of the solution vector x(t). 10

• Select One • Zero • Infinity • None

lim t→+∞

u(t), where u(t) = x(t) ‖x(t)‖

, is a unit vector in the direction of the solution vector x(t).

[Select One/U1/-U1/U2/-U2/None]

(d) Use the graph below to find lim t→−∞

‖x(t)‖.

• Select One • Zero • Infinity • None

(e) Use the graph below to find the lim t→−∞

u(t), where u(t) = x(t) ‖x(t)‖

[Select One/U1/-U1/U2/-U2/None]

(f) Characterize the zero solution, x0 = 0.

• Select One • Source Node • Source Spiral • Sink Node • Sink Spiral • Saddle • Center • None

x(t), and the associated unit vector u(t) = x(t)/‖x(t)‖. • You can move the time slider to see how each possible solution vector x(t) and unit vector u(t)

change in time. • You can move the eigenvectors v1 and v2 by dragging them from the endpoint, and then see how

the curves would change.

See in LN, § 5.2, See Examples 5.2.1, 5.2.2, 5.2.3, 5.2.4. 8. (10 points)

Note: You have only 5 attempts to solve this problem.

Match each vector field with its differential equation.

? 1. x′ = [

1 1 −1 1

] x

? 2. x′ = [

0 −1 1 0

] x

? 3. x′ = [

0 3 3 0

] x

? 4. x′ = [ −4 1 −1 −4

] x

A B

C D 12

See in LN, § 5.2, See Examples 5.2.1, 5.2.2, 5.2.3, 5.2.4. 9. (10 points)

Note: You have only 5 attempts to solve this problem.

Match each solution (in red) with its initial value problem.

? 1. x′ = [

0 3 3 0

] x, x(0) =

[ 0 −2

] ? 2. x′ =

[ 0 3 3 0

] x, x(0) =

[ −2 0

] ? 3. x′ =

[ 0 3 3 0

] x, x(0) =

[ 0 2

] ? 4. x′ =

[ 0 3 3 0

] x, x(0) =

[ 2 0

] 13

A B

C D

See in LN, § 5.2, See Examples 5.2.1, 5.2.2, 5.2.3, 5.2.4. 10. (10 points)

Note: You have only 5 attempts to solve this problem.

Match each solution (in red) with its initial value problem.

? 1. x′ = [

5 1 −1 5

] x, x(0) =

[ 2 4

] ? 2. x′ =

[ −5 1 −1 −5

] x, x(0) =

[ 2 4

] ? 3. x′ =

[ 5 1 −1 5

] x, x(0) =

[ 3 0

] ? 4. x′ =

[ −5 1 −1 −5

] x, x(0) =

[ 3 0

]

A B

C D

See in LN, § 5.2, See Examples 5.2.1, 5.2.2, 5.2.3, 5.2.4. 11. (10 points)

Note: You have only 5 attempts to solve this problem.

Match each solution (in red) with its initial value problem.

? 1. x′ = [

2 1 1 2

] x, x(0) =

[ 2 0

] ? 2. x′ =

[ −2 1 1 −2

] x, x(0) =

[ 0 2

] ? 3. x′ =

[ −2 1 1 −2

] x, x(0) =

[ 2 0

] ? 4. x′ =

[ 2 1 1 2

] x, x(0) =

[ 0 2

]

A B

C D 16

See in LN, § 5.2, See Examples 5.2.1, 5.2.2, 5.2.3, 5.2.4. 12. (10 points)

Note: You have only 5 attempts to solve this problem.

Match each solution (in red) with its initial value problem.

? 1. x′ = [

0 −3 −3 0

] x, x(0) =

[ 2 0

] ? 2. x′ =

[ 0 −3 −3 0

] x, x(0) =

[ 0 2

] ? 3. x′ =

[ 0 3 3 0

] x, x(0) =

[ 0 2

] ? 4. x′ =

[ 0 3 3 0

] x, x(0) =

[ 2 0

] 17

A B

C D