Underdamped harmonic oscillator

DE, StudentNumber: 92

Underdamped harmonic oscillator

Let

a= 0.6, b= 0.9, t₀ = 0, t₁ = 5.9, (1)

ic0 = 1.1, ic1=−0.7, (2)

y⁰⁰(t) +ay⁰(t) +by(t) =f(t). (3)

The first order form of (3) is d dt

y v

=A y

v

+B(f(t)) =

0 1

−b −a y v

+

0 1

(f(t)), (4)

wherey⁰=v. The initial condition (2) is

~ y(0) =

y(0) v(0)

= ic₀

ic₁

. (5)

1, A

Exercise. Let f(t) = 0, (2) be true. How much isy(t₁) ?

Answers. A: 0.132345 B: 0.163314 C: 0.20153 D: 0.248688 E: 0.306881 2, Aa

Exercise. Letf(t) = 0, (2) be true. LetUt1,t0(~z) =~y(t1), where~y(t) is the solution of (4) with the initial condition

~

y(t₀) =~z. How much is the first component of

Ut1,t0( ic₀

ic1

).

Answers. A: 0.163314 B: 0.20153 C: 0.248688 D: 0.306881 E: 0.378691 3, Ab

Exercise. Letf(t) = 0, (2) be true. LetUt1,t0(~z) =~y(t1), where~y(t) is the solution of (4) with the initial condition

~

y(t0) =~z. Compute Ut1,t0(~z), if~z=n1(0.2,0)^T +n2(0,0.2)^T, n1,2∈ {0,1,2,3,4,5}. Plot these points!

Answers.

4, Ac

Exercise. f(t) = 0, (2) is true. Solve the following algebro-differential equation!

y⁰=v, v⁰ =−spring−f riction, spring=by, f riction=av.

How much isspring(t₁) ?

Answers. A: 0.119111 B: 0.146983 C: 0.181377 D: 0.223819 E: 0.276193 5, B

Exercise. f(t) = 0. The general solution of (3) is

y(t) =C1e^λ1t+C2e^λ2t, =(λ₁)>0.

How much is=(C₁), if (2) is satisfied?

Answers. A: 0.109392 B: 0.134989 C: 0.166577 D: 0.205556 E: 0.253656 6, Ba

Exercise. f(t) = 0. The general solution of (4) is

~

y(t) =C1e^λ1t~v1+C2e^λ2t~v2, =(λ₁)>0, ~v1 = (1, s21)^T, ~v2 = (1, s22)^T How much is=(C₁), if (10) is satisfied?

Answers. A: 0.109392 B: 0.134989 C: 0.166577 D: 0.205556 E: 0.253656 7, C

Exercise. f(t) = 0. The general solution of (3) is

y(t) =e^αt(C₁cosωt+C₂sinωt).

8, D

Exercise. f(t) = 0, (2), (3) true. Plot y(t) !

Answers.

1 2 3 4 5 6

-0.5 0.5 1.0

1 2 3 4 5 6

-0.5 0.5 1.0

1 2 3 4 5 6

0.5 1.0

1 2 3 4 5 6

-0.5 0.5 1.0

9, E

Exercise. f(t) = 0, (2), (3) is true. Plot y(t), y⁰(t)-t on a single figure!

Answers.

1 2 3 4 5 6

-0.5 0.5 1.0

1 2 3 4 5 6

-0.5 0.5 1.0

1 2 3 4 5 6

-1.0 -0.5 0.5 1.0

1 2 3 4 5 6

-0.5 0.5 1.0

10, F

Exercise. f(t) = 0, (2), (3) are true. Plot the

γ : [t₀, t₁]→R², γ(t) = (y(t), v(t))^T parametric curve!

Answers.

-0.5 0.5 1.0

-1.0 -0.5 0.5

0.5 1.0

-0.6 -0.4 -0.2

-0.5 0.5 1.0

-0.8 -0.6 -0.4 -0.2 0.2

-0.5 0.5 1.0

-0.8 -0.6 -0.4 -0.2 0.2 0.4

11, G

Exercise. f(t) = 0, (2), (3) are true. Plot the

γ : [t₀, t₁]→R³, γ(t) = (t, y(t), v(t))^T parametric curve!

0

2

4

6 0.0

0.5 1.0---0.00.60.40.2

0

2

4

6 -0.5

0.0 0.5

1.0--1.00.5 0.0

0.5

0

2

4

6-0.5 0.0 0.5

1.0-0.5

0.0 0

2

4

6-0.5 0.0

0.51.0-0.5 0.0

12, H

Exercise. Plot the (y, v)^T →A(y, v)^T vector field!

Answers.

-1.0 -0.5 0.0 0.5 1.0 -1.0

-0.5 0.0 0.5 1.0

-1.0 -0.5 0.0 0.5 1.0 -1.0

-0.5 0.0 0.5 1.0

-1.0 -0.5 0.0 0.5 1.0 -1.0

-0.5 0.0 0.5 1.0

-1.0 -0.5 0.0 0.5 1.0 -1.0

-0.5 0.0 0.5 1.0

13, Ha

Exercise. f(t) = 0. Plot the (y, v)^T →A(y, v)^T vector field and a solution of (4)!

Answers.

-1.0 -0.5 0.0 0.5 1.0 -1.0

-0.5 0.0 0.5 1.0

-1.0 -0.5 0.0 0.5 1.0 -1.0

-0.5 0.0 0.5 1.0

-1.0 -0.5 0.0 0.5 1.0 -1.0

-0.5 0.0 0.5 1.0

-1.0 -0.5 0.0 0.5 1.0 -1.0

-0.5 0.0 0.5 1.0

14, I

Exercise. Compute U = exp(1.4A) ! How much isU11 ?

Answers. A: 0.217904 B: 0.268893 C: 0.331814 D: 0.409459 E: 0.505272 15, J

Exercise. Compute U(t) = exp(tA)-t and plot its first column’s functions!

1 2 3 4 5 6

-0.5 0.5 1.0

1 2 3 4 5 6

-0.5 0.5 1.0

1 2 3 4 5 6

-0.5 0.5 1.0

1 2 3 4 5 6

-0.4 -0.2 0.2 0.4 0.6 0.8 1.0

16, K

Exercise. f(t) = 0, (2) true, y(0) = 1, y⁰(0) = 0. Plot (y(t), y⁰(t))^T !

Answers.

1 2 3 4 5 6

-0.5 0.5 1.0

1 2 3 4 5 6

-0.5 0.5 1.0

1 2 3 4 5 6

-0.5 0.5 1.0

1 2 3 4 5 6

-0.4 -0.2 0.2 0.4 0.6 0.8 1.0

17, L

Exercise. Compute U(t) = exp(tA)-t and plot its second column’s functions!

Answers.

1 2 3 4 5 6

-0.5 0.5 1.0

1 2 3 4 5 6

-0.5 0.5 1.0

1 2 3 4 5 6

-0.4 -0.2 0.2 0.4 0.6 0.8 1.0

1 2 3 4 5 6

-0.4 -0.2 0.2 0.4 0.6 0.8 1.0

18, M

Exercise. f(t) = 0, (2) true, y(0) = 0, y⁰(0) = 1. Plot (y(t), y⁰(t))^T !

Answers.

1 2 3 4 5 6

-0.5 0.5 1.0

1 2 3 4 5 6

-0.4 -0.2 0.2 0.4 0.6 0.8 1.0

1 2 3 4 5 6

-0.4 -0.2 0.2 0.4 0.6 0.8 1.0

1 2 3 4 5 6

-0.5 0.5 1.0

Overdamped harmonic oscillator

Let

a= 0, b= 1.84, t0 = 0, t1 = 2.6, (6)

ic0= 0.7, ic1 = 1.4, (7)

y⁰⁰(t) +ay⁰(t) +by(t) =f(t). (8)

The first order form of (8) is d dt

y v

=A y

v

+B(f(t)) =

0 1

−b −a y v

+

0 1

(f(t)), (9)

wherey⁰=v. The initial condition (7) is

~ y(0) =

y(0) v(0)

= ic₀

ic₁

. (10)

19, A

Exercise. Let f(t) = 0, (2) be true. How much isy(t₁) ?

Answers. A: 0.162461 B: 0.200477 C: 0.247389 D: 0.305278 E: 0.376713 20, Aa

Exercise. Letf(t) = 0, (2) be true. LetUt1,t0(~z) =~y(t1), where~y(t) is the solution of (4) with the initial condition

~

y(t₀) =~z. How much is the first component of

Ut1,t0( ic₀

ic1

).

Answers. A: 0.106689 B: 0.131654 C: 0.162461 D: 0.200477 E: 0.247389

21, Ab

Exercise. Let f(t) = 0, (2) be true. Let U_t1/6,t0(~z) = ~y(t₁/6), where ~y(t) is the solution of (4) with the initial condition~y(t₀) =~z. ComputeU_t1/6,t0(~z), if~z=n₁(0.2,0)^T+n₂(0,0.2)^T, n_1,2∈ {0,1,2,3,4,5}. Plot these points!

Answers.

22, Ac

Exercise. f(t) = 0, (2) is true. Solve the following algebro-differential equation!

y⁰=v, v⁰ =−spring−f riction, spring=by, f riction=av.

How much isspring(t₁) ?

Answers. A: 0.298929 B: 0.368878 C: 0.455196 D: 0.561711 E: 0.693152 23, B

Exercise. f(t) = 0. The general solution of (3) is

y(t) =C1e^λ1t+C2e^λ2t, λ1 > λ2. How much isC1, if (2) is satisfied?

Answers. A: 0. B: 0. C: 0. D: 0. E: 0.

24, Ba

Exercise. f(t) = 0. The general solution of (4) is

~

y(t) =C₁e^λ1t~v₁+C₂e^λ2t~v₂, λ₁> λ₂, ~v₁ = (1, s₂₁)^T, ~v₂ = (1, s₂₂)^T How much is=(C₁), if (10) is satisfied?

Answers. A: 1.31779 B: 1.62615 C: 2.00667 D: 2.47623 E: 3.05566

25, D

Exercise. f(t) = 0, (2), (3) true. Plot y(t) !

Answers.

0.5 1.0 1.5 2.0 2.5

0.2 0.4 0.6 0.8 1.0

0.5 1.0 1.5 2.0 2.5

0.2 0.4 0.6 0.8

0.5 1.0 1.5 2.0 2.5 0.6

0.7 0.8 0.9 1.0 1.1

0.5 1.0 1.5 2.0 2.5

0.3 0.4 0.5 0.6 0.7 0.8 0.9

26, E

Exercise. f(t) = 0, (2), (3) is true. Plot y(t), y⁰(t)-t on a single figure!

Answers.

0.5 1.0 1.5 2.0 2.5

-0.5 0.5 1.0 1.5

0.5 1.0 1.5 2.0 2.5

0.5 1.0 1.5

0.5 1.0 1.5 2.0 2.5

-0.5 0.5 1.0 1.5

0.5 1.0 1.5 2.0 2.5

-0.5 0.5 1.0 1.5

27, F

Exercise. f(t) = 0, (2), (3) are true. Plot the

γ : [t₀, t₁]→R², γ(t) = (y(t), v(t))^T parametric curve!

Answers.

0.60.70.80.91.01.1 0.0

0.5 1.0 1.5

0.2 0.4 0.6 0.8

-0.6 -0.4 -0.2 0.2 0.4

0.4 0.5 0.6 0.7 0.8 0.9

-0.4 -0.2 0.2 0.4

0.2 0.4 0.6 0.8 1.0

-0.5 0.5 1.0 1.5

28, G

Exercise. f(t) = 0, (2), (3) are true. Plot the

γ : [t0, t1]→R³, γ(t) = (t, y(t), v(t))^T parametric curve!

Answers.

0

1

2

0.4 0.60.8--0.40.2

0.0 0.2 0.4

0

1 0.0 2

0.5 1.0

-0.5 0.0

0.5 1.0

0

1

2 0.60.81.0

0.0 0.5

1.0 0

1

2 0.00.20.40.60.8-0.5 0.0

29, H

Exercise. Plot the (y, v)^T →A(y, v)^T vector field!

Answers.

-1.0 -0.5 0.0 0.5 1.0 -1.0

-0.5 0.0 0.5 1.0

-1.0 -0.5 0.0 0.5 1.0 -1.0

-0.5 0.0 0.5 1.0

-1.0 -0.5 0.0 0.5 1.0 -1.0

-0.5 0.0 0.5 1.0

-1.0 -0.5 0.0 0.5 1.0 -1.0

-0.5 0.0 0.5 1.0

30, Ha

Exercise. f(t) = 0. Plot the (y, v)^T →A(y, v)^T vector field and a solution of (4)!

Answers.

-1.0 -0.5 0.0 0.5 1.0 -1.0

-0.5 0.0 0.5 1.0

-1.0 -0.5 0.0 0.5 1.0 -1.0

-0.5 0.0 0.5 1.0

-1.0 -0.5 0.0 0.5 1.0 -1.0

-0.5 0.0 0.5 1.0

-1.0 -0.5 0.0 0.5 1.0 -1.0

-0.5 0.0 0.5 1.0

31, I

Exercise. Compute U = exp(1.4A) ! How much isU11 ?

Answers. A: 0.206568 B: 0.254905 C: 0.314552 D: 0.388157 E: 0.478986 32, J

Exercise. Compute U(t) = exp(tA)-t and plot its first column’s functions!

0.5 1.0 1.5 2.0 2.5

-0.5 0.5 1.0

0.5 1.0 1.5 2.0 2.5

-0.5 0.5 1.0

0.5 1.0 1.5 2.0 2.5

-0.2 0.2 0.4 0.6 0.8 1.0

0.5 1.0 1.5 2.0 2.5

-0.5 0.5 1.0

33, K

Exercise. f(t) = 0, (2) true, y(0) = 1, y⁰(0) = 0. Plot (y(t), y⁰(t))^T !

Answers.

0.5 1.0 1.5 2.0 2.5

-0.2 0.2 0.4 0.6 0.8 1.0

0.5 1.0 1.5 2.0 2.5

-0.5 0.5 1.0

0.5 1.0 1.5 2.0 2.5

-0.5 0.5 1.0

0.5 1.0 1.5 2.0 2.5

-0.5 0.5 1.0

34, L

Exercise. Compute U(t) = exp(tA)-t and plot its second column’s functions!

Answers.

0.5 1.0 1.5 2.0 2.5

-0.2 0.2 0.4 0.6 0.8 1.0

0.5 1.0 1.5 2.0 2.5

0.2 0.4 0.6 0.8 1.0

0.5 1.0 1.5 2.0 2.5

-0.1 0.1 0.2 0.3 0.4 0.5

0.5 1.0 1.5 2.0 2.5

0.2 0.4 0.6 0.8 1.0

35, M

Exercise. f(t) = 0, (2) true, y(0) = 0, y⁰(0) = 1. Plot (y(t), y⁰(t))^T !

Answers.

0.5 1.0 1.5 2.0 2.5

0.2 0.4 0.6 0.8 1.0

0.5 1.0 1.5 2.0 2.5

0.2 0.4 0.6 0.8 1.0

0.5 1.0 1.5 2.0 2.5

-0.1 0.1 0.2 0.3 0.4 0.5

0.5 1.0 1.5 2.0 2.5

-0.2 0.2 0.4 0.6 0.8 1.0

Critical damping of a harmonic oscillator

Let

a= 2., b= 0.64, t0 = 2., t1= 2., (11)

ic0= 1., ic1 = 0.7, (12)

y⁰⁰(t) +ay⁰(t) +by(t) =f(t). (13)

The first order form of (13) is d dt

y v

=A y

v

+B(f(t)) =

0 1

−b −a y v

+

0 1

(f(t)), (14)

wherey⁰=v. The initial condition (12) is

~ y(0) =

y(0) v(0)

= ic₀

ic₁

. (15)

36, A

Exercise. Let f(t) = 0, (2) be true. How much isy(t₁) ?

Answers. A: 0.34828 B: 0.429777 C: 0.530345 D: 0.654446 E: 0.807586 37, Aa

Exercise. Letf(t) = 0, (2) be true. LetUt1,t0(~z) =~y(t1), where~y(t) is the solution of (4) with the initial condition

~

y(t₀) =~z. How much is the first component of

Ut1,t0( ic₀

ic1

).

Answers. A: 0.654446 B: 0.807586 C: 0.996561 D: 1.22976 E: 1.51752

38, Ab

Exercise. Let f(t) = 0, (2) be true. Let U_t1/6,t0(~z) = ~y(t₁/6), where ~y(t) is the solution of (4) with the initial condition~y(t₀) =~z. ComputeU_t1/6,t0(~z), if~z=n₁(0.2,0)^T+n₂(0,0.2)^T, n_1,2∈ {0,1,2,3,4,5}. Plot these points!

Answers.

39, Ac

Exercise. f(t) = 0, (2) is true. Solve the following algebro-differential equation!

y⁰=v, v⁰ =−spring−f riction, spring=by, f riction=av.

How much isspring(t1) ?

Answers. A: 0.275057 B: 0.339421 C: 0.418845 D: 0.516855 E: 0.637799 40, B

Exercise. f(t) = 0. The general solution of (3) is

y(t) =C₁e^λt+C₂te^λt. How much isC1, if (2) is satisfied?

Answers. A: -0.0656704 B: -0.0810373 C: -0.1 D: -0.1234 E: -0.152276 41, Ba

Exercise. f(t) = 0. The general solution of (4) is

~

y(t) =C₁e^λ1t~v₁+C₂e^λ2t~v₂, How much isC1(~v2)1, if (10) is satisfied?

Answers. A: -0.0656704 B: -0.0810373 C: -0.1 D: -0.1234 E: -0.152276

42, D

Exercise. f(t) = 0, (2), (3) true. Plot y(t) !

Answers.

0.5 1.0 1.5 2.0

1.05 1.10 1.15 1.20 1.25

0.5 1.0 1.5 2.0

0.9 1.0 1.1

0.5 1.0 1.5 2.0

0.7 0.8 0.9 1.0 1.1

0.5 1.0 1.5 2.0

0.95 1.00 1.05 1.10 1.15 1.20

43, E

Exercise. f(t) = 0, (2), (3) is true. Plot y(t), y⁰(t)-t on a single figure!

Answers.

0.5 1.0 1.5 2.0

0.5 1.0

0.5 1.0 1.5 2.0

0.5 1.0

0.5 1.0 1.5 2.0

0.5 1.0

0.5 1.0 1.5 2.0

0.5 1.0

44, F

Exercise. f(t) = 0, (2), (3) are true. Plot the

γ : [t₀, t₁]→R², γ(t) = (y(t), v(t))^T parametric curve!

Answers.

0.80.91.01.1

-0.4 -0.2 0.2 0.4 0.6

0.91.01.1

-0.4 -0.2 0.2 0.4 0.6

1.001.051.101.151.20 -0.2

0.2 0.4 0.6

1.101.151.201.25 -0.2

0.2 0.4 0.6

45, G

Exercise. f(t) = 0, (2), (3) are true. Plot the

γ : [t0, t1]→R³, γ(t) = (t, y(t), v(t))^T parametric curve!

Answers.

0.0 0.5

1.0

1.5

2.0

1.01.11.2 0.0

0.5

0.0 0.5

1.0

1.5

2.0

0.8 1.0

0.0 0.5

0.0 0.5

1.0

1.5

2.0

0.80.91.01.1 0.0

0.5 0.0

0.5

1.0

1.5

2.0

1.01.11.2 0.0

0.5

46, H

Exercise. Plot the (y, v)^T →A(y, v)^T vector field!

Answers.

-1.0 -0.5 0.0 0.5 1.0 -1.0

-0.5 0.0 0.5 1.0

-1.0 -0.5 0.0 0.5 1.0 -1.0

-0.5 0.0 0.5 1.0

-1.0 -0.5 0.0 0.5 1.0 -1.0

-0.5 0.0 0.5 1.0

-1.0 -0.5 0.0 0.5 1.0 -1.0

-0.5 0.0 0.5 1.0

47, Ha

Exercise. f(t) = 0. Plot the (y, v)^T →A(y, v)^T vector field and a solution of (4)!

Answers.

-1.0 -0.5 0.0 0.5 1.0 -1.0

-0.5 0.0 0.5 1.0

-1.0 -0.5 0.0 0.5 1.0 -1.0

-0.5 0.0 0.5 1.0

-1.0 -0.5 0.0 0.5 1.0 -1.0

-0.5 0.0 0.5 1.0

-1.0 -0.5 0.0 0.5 1.0 -1.0

-0.5 0.0 0.5 1.0

48, I

Exercise. Compute U = exp(1.4A) ! How much isU11 ?

Answers. A: 0.454251 B: 0.560546 C: 0.691713 D: 0.853574 E: 1.05331 49, J

Exercise. Compute U(t) = exp(tA)-t and plot its first column’s functions!

0.5 1.0 1.5 2.0

-0.4 -0.2 0.2 0.4 0.6 0.8 1.0

0.5 1.0 1.5 2.0

-0.2 0.2 0.4 0.6 0.8 1.0

0.5 1.0 1.5 2.0

-0.2 0.2 0.4 0.6 0.8 1.0

0.5 1.0 1.5 2.0

-0.2 0.2 0.4 0.6 0.8 1.0

50, K

Exercise. f(t) = 0, (2) true, y(0) = 1, y⁰(0) = 0. Plot (y(t), y⁰(t))^T !

Answers.

0.5 1.0 1.5 2.0

-0.2 0.2 0.4 0.6 0.8 1.0

0.5 1.0 1.5 2.0

-0.2 0.2 0.4 0.6 0.8 1.0

0.5 1.0 1.5 2.0

-0.4 -0.2 0.2 0.4 0.6 0.8 1.0

0.5 1.0 1.5 2.0

-0.2 0.2 0.4 0.6 0.8 1.0

51, L

Exercise. Compute U(t) = exp(tA)-t and plot its second column’s functions!

Answers.

0.5 1.0 1.5 2.0

0.2 0.4 0.6 0.8 1.0

0.5 1.0 1.5 2.0

0.2 0.4 0.6 0.8 1.0

0.5 1.0 1.5 2.0

0.2 0.4 0.6 0.8 1.0

0.5 1.0 1.5 2.0

0.2 0.4 0.6 0.8 1.0

52, M

Exercise. f(t) = 0, (2) true, y(0) = 0, y⁰(0) = 1. Plot (y(t), y⁰(t))^T !

Answers.

0.5 1.0 1.5 2.0

0.2 0.4 0.6 0.8 1.0

0.5 1.0 1.5 2.0

0.2 0.4 0.6 0.8 1.0

0.5 1.0 1.5 2.0

0.2 0.4 0.6 0.8 1.0

0.5 1.0 1.5 2.0

0.2 0.4 0.6 0.8 1.0

Distributions (δ, δ

)

LetχA(t) = 1, ift∈A, otherwise χA(t) is zero.

n= 1.15, a= 6, φ(t) =e^at, αn(t) =nχ[−1/(2n),1/(2n)](t), β_n(t) = 1

√2πσ_ne^{−t2/(2σn2)}, σ_n= 1/(4n),

˜

αn(t) =n²(χ_[−1/n,0](t)−χ_[0,1/n](t)), β˜n(t) =β_n⁰(t), hf, φi=

Z ∞

−∞

f(t)φ(t)dt 53, A

Exercise. Plot αn(t) and βn(t) !

Answers.

-1.0 -0.5 0.5 1.0

2 4 6 8 10

-1.0 -0.5 0.5 1.0

2 4 6 8

-1.0 -0.5 0.5 1.0

2 4 6 8 10 12

-1.0 -0.5 0.5 1.0

2 4 6 8 10 12 14

54, Aa

Answers.

-0.3 -0.2 -0.1 0.1 0.2 0.3 2

4 6 8 10 12 14

-0.3 -0.2 -0.1 0.1 0.2 0.3 5

10 15 20

-0.3 -0.2 -0.1 0.1 0.2 0.3 5

10 15

-0.3 -0.2 -0.1 0.1 0.2 0.3 5

10 15 20

55, B

Exercise. Plot αe_n(t) and βe_n(t)!

Answers.

-1.0 -0.5 0.5 1.0

-200 -100 100 200

-1.0 -0.5 0.5 1.0

-150 -100 -50 50 100 150

-1.0 -0.5 0.5 1.0

-300 -200 -100 100 200 300

-1.0 -0.5 0.5 1.0

-100 -50 50 100

56, C

Exercise. Compute φ(0)− hα, φi ?

Answers. A: -0.00100566 B: -0.00124098 C: -0.00153137 D: -0.00188972 E: -0.00233191 57, D

Exercise. Compute (−φ⁰(0))− hα,φie ?

Answers. A: 0.00231479 B: 0.00285645 C: 0.00352486 D: 0.00434967 E: 0.0053675 58, E

Exercise. Compute φ(0)− hβ, φi !

Answers. A: -0.000754331 B: -0.000930845 C: -0.00114866 D: -0.00141745 E: -0.00174913

59, F

Exercise. Compute (−φ(0))− hβ, φie !

Answers. A: 0.000702983 B: 0.000867481 C: 0.00107047 D: 0.00132096 E: 0.00163007 60, G

Exercise. Let

n= 2ⁱ, i= 0, . . . ,4, err_n=|φ(0)− hα_n, φi|.

Plot the points [ln(n),ln(errn)] and find the best fitting line by linear regression (i.e. by the method of least squares)!

Answers.

0.5 1.0 1.5 2.0 2.5

-8 -6 -4 -2

0.5 1.0 1.5 2.0 2.5

-8 -6 -4 -2

0.5 1.0 1.5 2.0 2.5

-8 -6 -4 -2

0.5 1.0 1.5 2.0 2.5

-8 -6 -4 -2

61, H Exercise. Let

n= 2ⁱ, i= 0, . . . ,4, err_n=|φ(0)− hα_n, φi|.

Take the points [ln(n),ln(errn)] and find the best fitting line by linear regression (i.e. by the method of least squares)! What is the slope of that line?

Answers. A: -0.86481 B: -1.06718 C: -1.31689 D: -1.62505 E: -2.00531 62, Hb

Exercise. Let

n= 2ⁱ, i= 0, . . . ,4, errn=|φ(0)− hγ_n, φi|, where

γn(t) =χ_[0,1/n](t)

Take the points [ln(n),ln(err_n)] and find the best fitting line by linear regression (i.e. by the method of least squares)! What is the slope of that line?

Answers. A: -0.921199 B: -1.13676 C: -1.40276 D: -1.73101 E: -2.13606 63, I

Exercise. Let

n= 2ⁱ, i= 0, . . . ,4, rr =|(−φ⁰(0))− hα , φi|.

Answers.

0.5 1.0 1.5 2.0 2.5

-8 -6 -4 -2

0.5 1.0 1.5 2.0 2.5

-8 -6 -4 -2

0.5 1.0 1.5 2.0 2.5

-8 -6 -4 -2

0.5 1.0 1.5 2.0 2.5

-8 -6 -4 -2

64, J

Exercise. Let

n= 2ⁱ, i= 2, . . . ,13, errn=|φ(0)− hα_n, φi|.

Take the points [ln(n),ln(errn)] and find the best fitting line by linear regression (i.e. by the method of least squares)! What is the slope of that line?

Answers. A: -2.01418 B: -2.4855 C: -3.06711 D: -3.78481 E: -4.67045 65, K

Exercise. y⁰(t) =δ(t), y(−1) = 0, y⁰_n(t) =αn(t), yn(−1) = 0.Plot y, yn-t!

Answers.

-0.4 -0.2 0.2 0.4 0.2

0.4 0.6 0.8 1.0

-0.4 -0.2 0.2 0.4 0.2

0.4 0.6 0.8 1.0

-0.4 -0.2 0.2 0.4 0.2

0.4 0.6 0.8 1.0

-0.4 -0.2 0.2 0.4 0.2

0.4 0.6 0.8 1.0

66, L

Exercise. y⁰(t) =δ(t), y(−1) = 0, y⁰_n(t) =β_n(t), y_n(−1) = 0.Plot y, y_n-t!

Answers.

-0.4 -0.2 0.2 0.4 0.2

0.4 0.6 0.8 1.0

-0.4 -0.2 0.2 0.4 0.2

0.4 0.6 0.8 1.0

-0.4 -0.2 0.2 0.4 0.2

0.4 0.6 0.8 1.0

-0.4 -0.2 0.2 0.4 0.2

0.4 0.6 0.8 1.0

67, M

Exercise. y⁰⁰(t) =δ(t), y(−1) = 0, y⁰(−1) = 0, y⁰⁰_n(t) =α_n(t), y_n(−1) = 0, y⁰(−1) = 0.Plot y, y_n-t!

Answers.

-0.2 -0.1 0.1 0.2

0.05 0.10 0.15 0.20

-0.2 -0.1 0.1 0.2

0.05 0.10 0.15 0.20

-0.2 -0.1 0.1 0.2

0.05 0.10 0.15 0.20

-0.2 -0.1 0.1 0.2

0.05 0.10 0.15 0.20

68, N

Exercise. y⁰⁰(t) =δ(t), y(−1) = 0, y⁰(−1) = 0, y⁰⁰_n(t) =βn(t), yn(−1) = 0, y⁰(−1) = 0.Plot y, yn-t!

Answers.

-0.2 -0.1 0.1 0.2

0.05 0.10 0.15 0.20

-0.2 -0.1 0.1 0.2

0.05 0.10 0.15 0.20

-0.2 -0.1 0.1 0.2

0.05 0.10 0.15 0.20

-0.2 -0.1 0.1 0.2

0.05 0.10 0.15 0.20

69, O

Exercise. y⁰⁰(t) =δ⁰(t), y(−1) = 0, y⁰(−1) = 0, y⁰⁰_n(t) =βe_n(t), y_n(−1) = 0, y⁰(−1) = 0.Plot y, y_n-t!

Answers.

-0.2 -0.1 0.1 0.2

0.2 0.4 0.6 0.8 1.0

-0.2 -0.1 0.1 0.2

0.2 0.4 0.6 0.8 1.0

-0.2 -0.1 0.1 0.2

0.2 0.4 0.6 0.8 1.0

-0.2 -0.1 0.1 0.2

0.2 0.4 0.6 0.8 1.0

70, P

Exercise. y⁰⁰(t) = 0.4δ(t) + 1.6δ⁰(t), y(−1) = 0, y⁰(−1) = 0, y_n⁰⁰(t) = 0.4β(t) + 1.6βen(t), yn(−1) = 0, y⁰(−1) = 0.

Ploty, y_n-t!

Answers.

-0.3 -0.2 -0.1 0.1 0.2 0.3 0.5

1.0 1.5

-0.3 -0.2 -0.1 0.1 0.2 0.3 0.5

1.0 1.5

-0.3 -0.2 -0.1 0.1 0.2 0.3 0.5

1.0 1.5

-0.3 -0.2 -0.1 0.1 0.2 0.3 0.5

1.0 1.5

71, Q

Exercise. y⁰⁰(t) = −3y(t)−y⁰(t) +δ(t), y(−1) = 0, y⁰(−1) = 0, y_n⁰⁰(t) = −3y_n(t)−y_n⁰(t) +β_n(t), y_n(−1) = 0, y⁰_n(−1) = 0.Plot y, yn-t!

Answers.

-0.15-0.10-0.05 0.05 0.10 0.15 0.02

0.04 0.06 0.08 0.10 0.12 0.14

-0.15-0.10-0.05 0.05 0.10 0.15 0.02

0.04 0.06 0.08 0.10 0.12 0.14

-0.15-0.10-0.05 0.05 0.10 0.15 0.02

0.04 0.06 0.08 0.10 0.12 0.14

-0.15-0.10-0.05 0.05 0.10 0.15 0.02

0.04 0.06 0.08 0.10 0.12 0.14

72, R

Exercise. y⁰(t) =−3y(t) +δ(t), y(−1) = 0, y_n⁰(t) =−3y_n(t) +β_n(t), y_n(−1) = 0.Plot y, y_n-t!

Answers.

-0.2 -0.1 0.1 0.2 0.2

0.4 0.6 0.8 1.0

-0.2 -0.1 0.1 0.2 0.2

0.4 0.6 0.8 1.0

-0.2 -0.1 0.1 0.2 0.2

0.4 0.6 0.8 1.0

-0.2 -0.1 0.1 0.2 0.2

0.4 0.6 0.8 1.0

73, S

Exercise. y⁰(t) =−3y(t) +δ(t), y(−1) = 0, y_n⁰(t) =−3y_n(t) +β_n(t), y_n(−1) = 0.Plot y, y_n-t!

Answers.

-0.2 -0.1 0.1 0.2 0.2

0.4 0.6 0.8 1.0

-0.2 -0.1 0.1 0.2 0.2

0.4 0.6 0.8 1.0

-0.2 -0.1 0.1 0.2 0.2

0.4 0.6 0.8 1.0

-0.2 -0.1 0.1 0.2 0.2

0.4 0.6 0.8 1.0

StudentNumber: 92

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

B A D B D D E B B C C A C D B A D B C E

21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

D C D C D C C A B C E B D D A E B C D C

41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

C B B B C C A C C D C C A C B C C C D B

61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80

E B C A A C C D C C D C D