StdNum: 20; DE, Teszt 1, Majus 9.

StdNum: 20; DE, Teszt 1, Majus 9.

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

Let d

dty= (y−5)(y−2)(y−1)², y(0) = 2.5.

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

MCQ. A: 6 B: 7 C: 8 D: 9 E: 10

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: 0 C: 1 D: 2 E: 3

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:∞ B: 1 C: 5 D: 2 E:−∞

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: 5 C: −∞ D:∞ E: 1

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

MCQ.

-0.2 0.0 0.2 0.4 0.6 1

2 3 4 5

-0.2 0.0 0.2 0.4 0.6 1

2 3 4 5

-0.2 0.0 0.2 0.4 0.6 1

2 3 4 5

-0.2 0.0 0.2 0.4 0.6 1

2 3 4 5

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

Let d

dty= (y−5)(y−2)(y−1)², y(0) = 2.5.

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

MCQ. A: 6 B: 7 C: 8 D: 9 E: 10

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: -1 B: 0 C: 1 D: 2 E: 3

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: 1 C: 5 D: 2 E:−∞

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: 5 C: −∞ D:∞ E: 1

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

MCQ.

-0.2 0.0 0.2 0.4 0.6 1

2 3 4 5

-0.2 0.0 0.2 0.4 0.6 1

2 3 4 5

-0.2 0.0 0.2 0.4 0.6 1

2 3 4 5

-0.2 0.0 0.2 0.4 0.6 1

2 3 4 5

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

Let

d dt~y=

f₁ f2

=

−(y₁+ 3)y₂ y1−1

.

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: -2 C: -1 D: 0 E: 1

Exercise (4). Hom.Lin. DE. Radioactiv decay: I →II →. . . Let

A=

−4 0 4 −6

, d

dt~y =A~y, ~y(0) = 5

4

.

Subexercise (A). Find the smallest eigenvalue ofA ! MCQ. A: -10 B: -9 C: -8 D: -7 E: -6

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:−¹₂ B: ¹₂ C: ³₂ D: ⁵₂ E: ⁷₂

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:−9 B:−8 C: −6 D:−4 E: −3

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

21 ? MCQ. A: 0.00550384 B: 0.00618301 C: 0.00694599 D: 0.00780313 E: 0.00876603 Exercise (5). Hom.Lin. DE. Overdamped oscillator

Let

¨

y+ay˙+by= 0, A=

0 1

−24 −10

, d

dt~y=A~y, ~y(0) = 5

4

.

Subexercise (A). Find the smallest eigenvalue ofA ! MCQ. A: -10 B: -9 C: -8 D: -7 E: -6

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

MCQ. A:−12 B: −11 C: −10 D:−8 E: −6

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:−15 B: −14 C: −12 D:−10 E: −9

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

21 ? MCQ. A: -0.033023 B: -0.0370981 C: -0.041676 D: -0.0468188 E: -0.0525962

Exercise (6). 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 (B). Constant accerelation.

d³y

dt³ = 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(1.419A))₁₃ ?

MCQ. A: 0.71012 B: 0.797748 C: 0.896191 D: 1.00678 E: 1.13102 Subexercise (C). Radioactive decay, I →II →. . ..

d dt~y=

−2 0 2 −2

~ y=A~y.

(Itt~y= (y_I, y_II)^T.) How much is (exp(1.419A))21 ?

MCQ. A: 0.131648 B: 0.147894 C: 0.166144 D: 0.186646 E: 0.209678 Exercise (7). 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 (C).

G⁰(t) + 3G(t) =δ(t), G(−1) = 0.

How much isG(0.3) ?

MCQ. A: 0.286769 B: 0.322156 C: 0.36191 D: 0.40657 E: 0.45674 Subexercise (E).

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

How much isG(0.3) ?

Exercise (8). Laplace transform Subexercise (A). Let

y⁰(t) + 5y(t) = 4t−2, y(0) = 4.

How much isY(1.) ?

MCQ. A: 0.792376 B: 0.890155 C: 1. D: 1.1234 E: 1.26203 Subexercise (B). Let

y⁰(t) + 5y(t) = 4t−2, y(0) = 4.

If

Y(s) =X

k,n

A_k,n

(s−αk)ⁿ, A_k,n6= 0, α₁> α₂ >· · · , then how much isA1,1 ?

MCQ. A:−¹⁸₂₅ B: −¹⁷₂₅ C: −¹⁶₂₅ D: −¹⁴₂₅ E:−¹²₂₅ Subexercise (C). Let

d dt~y =

−6 0 6 −5

~ y+

1 t

, ~y(0) = 7

4

.

How much isY1(1.) ?

MCQ. A:{1.01732} B: {1.14286} C: {1.28389}D: {1.44232}E:{1.6203}

Exercise (9). Fourier transform, heat equation.

Subexercise (A). Letχ_[a,b](x) = 1, if x∈[a, b], otherwise it is zero. If χ_[0.7,2.2](x) =

∞

X

n=−∞

ˆ

χnen(x), ahol en(x) = e^inx

√2π on the (−π, π) interval, then how much is|ˆχ₂|?

MCQ. A: 0.397943 B: 0.447049 C: 0.502215 D: 0.564188 E: 0.633809 Subexercise (B). Let

φt(t, x) = 3φxx(t, x), φ(t, x) =φ(t, x+ 2π), φ(0, x) =χ_[0.7,2.2](x), ha x∈[−π, π].

If

φ(t, x) =

∞

X

n=−∞

c_n(t)e_n(x), Then how much is |c₂(0.2)|?

MCQ. A: 0.0361006 B: 0.0405554 C: 0.0455599 D: 0.051182 E: 0.0574979 Exercise (10). Euler-Lagrange equation, numerical methods.

Subexercise (D). (Heun’s method) Let

f(t, y) = (2 + 3t)(3 + 2y) t₀ = 2,∆t= 0.01.

d

dty(t) =f(t, y(t)), y(t0) =y0 = 3.

What is Heun’s prediction for y(t₀+ ∆t)-re?

MCQ. A: 2.99452 B: 3.36404 C: 3.77917 D: 4.24552 E: 4.76941

Subexercise (E). (Taylor series) Let

f(t, y) = (2 + 3t)(3 + 2y) t₀ = 2, d

dty(t) =f(t, y(t)), y(t0) =y0 = 3,

y(t0+ ∆t) =y0+c1∆t+c2∆t²+c3∆t³+O(∆t⁴).

How much isc₂ ?

MCQ. A: ¹¹⁷¹₂ B: ¹¹⁷³₂ C: ¹¹⁷⁵₂ D: ¹¹⁷⁷₂ E: ¹¹⁷⁹₂