Find the sum of the fixedpoints of the DE! MCQ

22

Exercise (1). Time-independent DE, one dimension. Qualitative behaviour, linearization.

Let d

dty= (y−3)(y−2)(y−1), y(0) = 1.5.

Subexercise (A). Find the sum of the fixedpoints of the DE!

MCQ. A: 4 B: 5 C: 6 D: 7 E: 8

Subexercise (B). Write down the linear approximation of the DE around the smallestyf ix fixedpoint as _dt^d∆y= a∆y ! How much isa?

MCQ. A: 1 B: 2 C: 3 D: 4 E: 5

Subexercise (C). With the given initial condition fory(0), how much is limt→∞y(t) ? (If limt→tcy(t) =±∞for somet_c>0, then answer±∞.)

MCQ. A: 3 B: ∞ C: 1 D:−∞ E: 2

Subexercise (D). With the given initial condition for y(0), how much is limt→−∞y(t) ? (If limt→tcy(t) =±∞

for somet_c<0, then answer±∞.) MCQ. A: 2 B: ∞ C: 3 D: 1 E:−∞

Subexercise (E). Plot the t→y(t) solutions of the DE!

MCQ.

-0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.5

1.0 1.5 2.0 2.5 3.0 3.5

-0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.5

1.0 1.5 2.0 2.5 3.0 3.5

-0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.5

1.0 1.5 2.0 2.5 3.0 3.5

-0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.5

1.0 1.5 2.0 2.5 3.0 3.5

Exercise (2). Time-independent DE, one dimension. Qualitative behaviour, linearization.

Let d

dty= (y−2)(y−1), y(0) = 1.5.

Subexercise (A). Find the sum of the fixedpoints of the DE!

MCQ. A: 1 B: 2 C: 3 D: 4 E: 5

Subexercise (B). Write down the linear approximation of the DE around the smallesty_{f ix} fixedpoint as _dt^d∆y= a∆y ! How much isa?

MCQ. A: -2 B: -1 C: 0 D: 1 E: 2

Subexercise (C). With the given initial condition fory(0), how much is limt→∞y(t) ? (If limt→t_cy(t) =±∞for sometc>0, then answer±∞.)

MCQ. A:−∞ B: ∞C: 2 D: 1

Subexercise (D). With the given initial condition for y(0), how much is limt→−∞y(t) ? (If limt→tcy(t) =±∞

for somet_c<0, then answer±∞.) MCQ. A: 1 B: 2 C: ∞ D:−∞

Subexercise (E). Plot the t→y(t) solutions of the DE!

MCQ.

0.0 0.2 0.4 0.6 0.8 1.0 0.5

1.0 1.5 2.0 2.5

0.0 0.2 0.4 0.6 0.8 1.0 0.5

1.0 1.5 2.0 2.5

0.0 0.2 0.4 0.6 0.8 1.0 0.5

1.0 1.5 2.0 2.5

0.0 0.2 0.4 0.6 0.8 1.0 0.5

1.0 1.5 2.0 2.5

Exercise (3). Time-independent DE, two dimensions. Qualitative behaviour, linearization.

Let

d dt~y =

f₁ f2

=

(y₁−1) (y₂−3) (y1−4) (y2−1)

.

Subexercise (A). Find the sum of the coordinates of the smallest~y_{f ix} fixed point!

MCQ. A: 0 B: 1 C: 2 D: 3 E: 4

Subexercise (B). Write down the linear approximation of the DE around the smallest ~yf ix as _dt^d∆~y = A∆~y ! How much is the sum of the elements ofA ?

MCQ. A: -6 B: -5 C: -4 D: -3 E: -2

Subexercise (C). Plot the phase portrait of the DE!

MCQ. ^{0 1 2 3 4 5}

0 1 2 3 4

Exercise (4). Time-independent DE, two dimensions. Qualitative behaviour, linearization.

Let

d dt~y =

f1

f₂

=

(y1−1) (y2−3) (y₁−4) (y₂−3)

.

Subexercise (A). Find the sum of the coordinates of that~yf ix fixed point where y1 = 1 ! MCQ. A: 2 B: 3 C: 4 D: 5 E: 6

Subexercise (B). Write down the linear approximation of the DE around the smallest ~yf ix as _dt^d∆~y = A∆~y ! How much is the sum of the elements ofA ?

MCQ. A: -4 B: -3 C: -2 D: -1 E: 0

Subexercise (C). Plot the phase portrait of the DE!

MCQ. ^{1 2 3 4}

1 2 3 4 5

Exercise (5). Time-independent DE, two dimensions. Qualitative behaviour, linearization.

Let

d dt~y =

f1

f2

=

(y1−1) (y2−3) y1−4

.

Subexercise (A). Find the sum of the coordinates of the~y_{f ix} fixed point!

MCQ. A: 5 B: 6 C: 7 D: 8 E: 9

Subexercise (B). Write down the linear approximation of the DE around the~yf ix as _dt^d∆~y=A∆~y ! How much is the sum of the elements ofA ?

MCQ. A: 3 B: 4 C: 5 D: 6 E: 7

Subexercise (C). Plot the phase portrait of the DE!

MCQ.

3.4 3.6 3.8 4.0 4.2 4.4 4.6 2.4

2.6 2.8 3.0 3.2 3.4 3.6

3.4 3.6 3.8 4.0 4.2 4.4 4.6 2.4

2.6 2.8 3.0 3.2 3.4 3.6

3.4 3.6 3.8 4.0 4.2 4.4 4.6 2.4

2.6 2.8 3.0 3.2 3.4 3.6

3.4 3.6 3.8 4.0 4.2 4.4 4.6 2.4

2.6 2.8 3.0 3.2 3.4 3.6

Exercise (6). Hom.Lin. DE. Radioactiv decay: I →II →. . . Let

A=

−2 0 2 −3

, d

dt~y =A~y, ~y(0) = 2

6

.

Subexercise (A). Find the smallest eigenvalue ofA ! MCQ. A: -5 B: -4 C: -3 D: -2 E: -1

Subexercise (B). Find the~v1 and~v2 eigenvectors corresponding to the λ1< λ2 eigenvalues. Normalize them by the condition (~vi)2 = 1, i= 1,2.Assemble them into a matrixS = (~v1, ~v2) matrixot. How much is the sum of the elements ofS ?

MCQ. A: ³₂ B: ⁵₂ C: ⁷₂ D: ⁹₂ E: ¹¹₂

Subexercise (C). The solution of the DE can be written as

~ y(t) =

2

X

i=1

Cie^λit~vi.

How much isC₁, if~y(0) satisfies the given initial condition?

MCQ. A:−1 B: 0 C: 2 D: 4 E: 5

Subexercise (D). Compute the matrix exponentiale^tA ! How much is e^1.4·A

21 ? MCQ. A: 0.0575301 B: 0.0646293 C: 0.0726046 D: 0.081564 E: 0.091629

Exercise (7). Hom.Lin. DE. Overdamped oscillator Let

¨

y+ay˙+by= 0, A=

0 1

−6 −5

, d

dt~y=A~y, ~y(0) = 2

6

.

Subexercise (A). Find the smallest eigenvalue ofA ! MCQ. A: -5 B: -4 C: -3 D: -2 E: -1

Subexercise (B). Find the~v₁ and~v₂ eigenvectors corresponding to the λ₁< λ₂ eigenvalues. Normalize them by the condition (~vi)1 = 1, i= 1,2.Assemble them into a matrixS = (~v1, ~v2) matrixot. How much is the sum of the elements ofS ?

MCQ. A:−5 B:−3 C: −1 D: 0 E: 1

Subexercise (C). The solution of the DE can be written as

~ y(t) =

2

X

i=1

C_ie^λit~v_i.

How much isC₁, if~y(0) satisfies the given initial condition?

MCQ. A:−13 B: −12 C: −10 D:−8 E: −7

Subexercise (D). Compute the matrix exponentiale^tA ! How much is e^1.4·A

21 ? MCQ. A: -0.17259 B: -0.193888 C: -0.217814 D: -0.244692 E: -0.274887

Exercise (8). Hom.Lin. DE. Underdamped oscillator.

Let

¨

y+ay˙+by= 0, A=

0 1

−5 −2

, d

dt~y=A~y, ~y(0) = 2

6

.

Subexercise (A). Find the absolute value of the imaginary part of the eigenvalues ofA ! MCQ. A:−4 B:−3 C: −1 D: 1 E: 2

Subexercise (B). Find the~v1 and~v2 eigenvectors corresponding to the λ1< λ2 eigenvalues. Normalize them by the condition (~vi)1 = 1, i= 1,2.Assemble them into a matrixS = (~v1, ~v2) matrixot. How much is the sum of the real parts of the elements ofS ?

MCQ. A:−2 B: 0 C: 2 D: 3 E: 4

Subexercise (C). The solution of the DE can be written as

~ y(t) =

2

X

i=1

C_ie^λit~v_i.

How much isC₁, if~y(0) satisfies the given initial condition?

MCQ. A:−2 B:−1 C: 1 D: 3 E: 4

Subexercise (D). Compute the matrix exponentiale^tA ! How much is e^1.4·A

21 ? MCQ. A: -0.129664 B: -0.145665 C: -0.16364 D: -0.183833 E: -0.206518

Exercise (9). Hom.Lin. DE. Jordan decomposition.

Theorem: For any complex square A matrix there exists an S such that SAS⁻¹ is block diagonal with blocks choosen from the following list

J₁ = λ

, J₂ = λ 1

0 λ

, J₃=





λ 1 0 0 λ 1 0 0 λ



, . . . .

Subexercise (A). Constant velocity motion.

¨

y= 0, d dt~y=

0 1 0 0

~ y=A~y.

(Here~y= (y,y)˙ ^T.) How much is (exp(0.9A))₁₂?

MCQ. A: 0.713138 B: 0.801139 C: 0.9 D: 1.01106 E: 1.13582 Subexercise (B). Constant accerelation.

d³y

d³ = 0, d dt~y =





0 1 0 0 0 1 0 0 0



~y=A~y.

(Here~y= (y,y,˙ y)¨^T.) How much is (exp(0.9A))13 ?

MCQ. A: 0.360513 B: 0.405 C: 0.454977 D: 0.511121 E: 0.574194 Subexercise (C). Radioactive decay, I →II →. . ..

d dt~y=

−2 0 3 −2

~ y=A~y.

(Itt~y= (y_I, y_II)^T.) How much is (exp(0.9A))21 ?

MCQ. A: 0.353643 B: 0.397282 C: 0.446307 D: 0.501381 E: 0.563252 Subexercise (D). Critically damped oscillator. Leta= 3,

¨

y+ 2ay˙+a²y= 0, d dt~y=

0 1

−a² −2a

~ y=A~y.

(Here~y= (y,y)˙ ^T.) How much is (exp(0.2·A))₁₂ ?

MCQ. A: 0.0689153 B: 0.0774194 C: 0.086973 D: 0.0977055 E: 0.109762 Exercise (10). Impulse response, distributions.

Dirac-delta: δ(t) = 0, hat6= 0, Z ∞

−∞

δ(t)dt= 1.

Heaviside theta: θ(t) =

(0, hat <0, 1, hat >0,

????: K(t) =

(0, hat <0, t, hat >0., hf(t), φ(t)i=

Z −∞

∞

f(t)φ(t)dt hf⁰(t), φ(t)i=

Z −∞

∞

f⁰(t)φ(t)dt=− Z −∞

∞

f(t)φ⁰(t)dt=−hf(t), φ⁰(t)i Then

θ⁰(t) =δ(t),

K⁰(t) =θ(t), K⁰⁰(t) =θ⁰(t) =δ(t).

Subexercise (A). Ifφ(t) = (1 +t²)⁻¹, then how much is

−hθ(t), φ⁰(t)i.

MCQ. A: -1 B: 0 C: 1 D: 2 E: 3 Subexercise (B).

1

3G⁰(t) =δ(t), G(−1) = 0.

How much isG(0.3) ?

MCQ. A: 2 B: 3 C: 4 D: 5 E: 6

Subexercise (C).

G⁰(t) + 3G(t) =δ(t), G(−1) = 0.

How much isG(0.3) ?

MCQ. A: 0.322156 B: 0.36191 C: 0.40657 D: 0.45674 E: 0.513102 Subexercise (D).

G⁰(t) + 3G(t) =δ(t), G(1) = 0.

How much isG(0.3) ?

MCQ. A: -0.4936 B: -0.3702 C: -0.2468 D: -0.1234 E: 0.

Subexercise (E).

G⁰⁰(t) + 3G(t) =δ(t), G(−1) = 0, G⁰(−1) = 0.

How much isG(0.3) ?

MCQ. A: 0.393451 B: 0.442003 C: 0.496546 D: 0.55782 E: 0.626655 Subexercise (F).

G⁰⁰(t) + 5G⁰(t) + 6G(t) =δ(t), G(−1) = 0G⁰(−1) = 0.

How much isG(0.3) ?

MCQ. A: 0.100329 B: 0.112709 C: 0.126617 D: 0.142242 E: 0.159795 Subexercise (G).

G⁰⁰(t) + 2G⁰(t) + 5G(t) =δ(t), G(−1) = 0G⁰(−1) = 0.

How much isG(0.3) ?

MCQ. A: 0.209149 B: 0.234958 C: 0.263951 D: 0.296523 E: 0.333114

Subexercise (H). Have you managed to plot the retarded Green function solutions of the following DE? (Just answer ”A: Yes”.)

G⁰+G=δ, G⁰⁰+G=δ,

G⁰⁰+ 3G⁰+ 2G=δ, G⁰⁰+ 2G⁰+ 5G=δ, MCQ. A: Igen/Yes

Exercise (11). Inhom. Lin. DE.

Let

χ_[a,b](t) =

(1, ha t∈[a, b], 0, ha t6∈[a, b].

Subexercise (A). Leta= 2.

G⁰(t) +aG(t) =δ(t), G(−1) = 0, G(t) =θ(t)e^−at. UseG to obtain the solution of the DE

y⁰(t) +ay(t) =χ_[1,2](t), y(−∞) = 0 fort >2-re! How much isy(3) ?

MCQ. A: 0.0463618 B: 0.0520828 C: 0.0585098 D: 0.0657299 E: 0.073841 Subexercise (B). Leta= 2.

G⁰(t) +aG(t) =δ(t), G(t) =θ(t)e^−at. UseG to obtain the solution of the DE

y⁰(t) +ay(t) =χ_[1,2](t), y(0) = 7 fort >2-ra! How much isy(3) ?

MCQ. A: 0.0675281 B: 0.0758611 C: 0.0852223 D: 0.0957388 E: 0.107553 Subexercise (C). Leta= 2

G⁰(t) +aG(t) =δ(t), G(−1) = 0, G(t) =θ(t)e^−at. UseG to obtain the solution of the DE

y⁰(t) +ay(t) =θ(t), y(0) = 0

fort >0 ! (That is the unit step response.) How much time is necessary fory to reach the 90 percent of the value of y(∞) ? (This is the rise time of the system. It might be defined as the time between the 10 and 90 percent values ofy(∞).)

0.0 0.5 1.0 1.5 2.0 2.5

0.1 0.2 0.3 0.4 0.5 0.6

MCQ. A: 0.912256 B: 1.02483 C: 1.15129 D: 1.29336 E: 1.45296 Subexercise (D). Leta= 2

G⁰(t) +aG(t) =δ(t), G(−1) = 0, G(t) =θ(t)e^−at. UseG to obtain the solution of the DE

y⁰(t) +ay(t) =θ(t), y(0) = 0

fort >0 ! What is the settling time of the system, i.e. where is that time moment after which the the value of y will stay in the ±2% neighbourhood of y(∞) ?

MCQ. A: 1.2281 B: 1.37965 C: 1.5499 D: 1.74115 E: 1.95601 Exercise (12). Inhom. Lin. DE.

Let

χ_[a,b](t) =

(1, ha t∈[a, b], 0, ha t6∈[a, b].

Subexercise (A). Lett₁= 3., A=

−2 0 2 −3

, e^tA=

e^−2t 0

−2e^−3t+ 2e^−2t e^−3t

.

Usee^tA to obtain the solution of the DE d

dt~y(t) =A~y(t) + 0

1

(χ_[1,2](t)), ~y(−∞) = 0

0

fort >2 ! How much is~y(3.)₂ ?

MCQ. A:{0.0124953}B: {0.0140372} C: {0.0157694}D: {0.0177154} E:{0.0199015}

Subexercise (B). Lett1 = 3., A=

−2 0 2 −3

, e^tA=

e^−2t 0

−2e^−3t+ 2e^−2t e^−3t

.

Usee^tA to obtain the solution of the following DE d

dt~y(t) =A~y(t) + 0

1

(χ_[1,2](t)), ~y(0) = 7

8

.

How much is~y(3.)₂ ?

MCQ. A:{0.0442687}B: {0.0497315} C: {0.0558684}D: {0.0627625} E:{0.0705074}