StdNum: 12; DE, Teszt 1, Majus 9.

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StdNum: 12; DE, Teszt 1, Majus 9.

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

Let d

dty= (y−4)(y−1)², y(0) = 3.5.

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

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

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: -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→tcy(t) =±∞for somet_c>0, then answer±∞.)

MCQ. A:−∞ B: 4 C: 1 D:∞

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:−∞ B: 4 C:∞ D: 1

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

MCQ.

-0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1

2 3 4

-0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1

2 3 4

-0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1

2 3 4

-0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1

2 3 4

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Exercise (2). Time-independent DE, two dimensions. Qualitative behaviour, linearization.

Let

d dt~y =

f₁ f2

=

(y₁−1) (y₂−2) (y1−2) (y2−1)

.

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

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: -5 B: -4 C: -3 D: -2 E: -1

Exercise (3). Hom.Lin. DE. Radioactiv decay: I →II →. . . Let

A=

−1 0 1 −4

, d

dt~y =A~y, ~y(0) = 4

1

.

Subexercise (A). Find the smallest eigenvalue ofA ! MCQ. A: -7 B: -6 C: -5 D: -4 E: -3

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

MCQ. A: 2 B: 3 C: 5 D: 7 E: 8

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

~ y(t) =

2

X

i=1

Cie^λⁱ^t~vi.

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

MCQ. A:−⁷₃ B:−⁵₃ C: −⁴₃ D: −¹₃ E: ¹₃

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

21 ? MCQ. A: 0.0508355 B: 0.0571086 C: 0.0641558 D: 0.0720726 E: 0.0809664

Exercise (4). 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

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(1.469A))13 ?

MCQ. A: 0.854958 B: 0.96046 C: 1.07898 D: 1.21213 E: 1.3617 Subexercise (C). Radioactive decay, I →II →. . ..

d dt~y=

−3 0 3 −3

~ y=A~y.

(Itt~y= (y_I, y_II)^T.) How much is (exp(1.469A))₂₁ ?

MCQ. A: 0.0378969 B: 0.0425734 C: 0.047827 D: 0.0537288 E: 0.060359 2

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

How much isG(0.3) ?

MCQ. A: 0.165299 B: 0.185697 C: 0.208612 D: 0.234354 E: 0.263274 Subexercise (E).

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

How much isG(0.3) ?

MCQ. A: 0.243653 B: 0.27372 C: 0.307497 D: 0.345442 E: 0.388069 Exercise (6). Laplace transform

Subexercise (A). Let

y⁰(t) + 5y(t) = 3t, y(0) = 6.

How much isY(1.2) ?

MCQ. A: 1.30376 B: 1.46465 C: 1.64539 D: 1.84843 E: 2.07652 Subexercise (B). Let

y⁰(t) + 5y(t) = 3t, y(0) = 6.

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:−₂₅³ B: −₂₅¹ C: ₂₅¹ D: ₂₅² E: ₂₅³ Subexercise (C). Let

d dt~y =

−5 0 5 −4

~ y+

1 t

, ~y(0) = 5

6

.

How much isY1(1.2) ?

MCQ. A:{0.745515} B: {0.837511}C: {0.94086} D:{1.05696} E:{1.18739}

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Exercise (7). Fourier transform, heat equation.

Subexercise (A). Letχ_[a,b](x) = 1, if x∈[a, b], otherwise it is zero. If χ_[0.4,1.2](x) =

∞

X

n=−∞

ˆ

χnen(x), ahol en(x) = e^inx

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

MCQ. A: 0.254748 B: 0.286184 C: 0.321499 D: 0.361172 E: 0.40574 Subexercise (B). Let

φt(t, x) = 3φxx(t, x), φ(t, x) =φ(t, x+ 2π), φ(0, x) =χ_[0.4,1.2](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: 0.0205717 B: 0.0231102 C: 0.025962 D: 0.0291657 E: 0.0327648 Exercise (8). Euler-Lagrange equation, numerical methods.

Subexercise (D). (Heun’s method) Let

f(t, y) = (2 + 4t)(4 + 2y) t₀ = 2,∆t= 0.01.

d

dty(t) =f(t, y(t)), y(t0) =y0 = 4.

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

MCQ. A: 4.73819 B: 5.32288 C: 5.97972 D: 6.71762 E: 7.54658

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