StdNum: 13; DE, Teszt 1, Majus 9.
Exercise (1). Time-independent DE, one dimension. Qualitative behaviour, linearization.
Let d
dty=−(y+ 1)2(y+ 2), y(0) =−1.5.
Subexercise (A). Find the sum of the fixedpoints of the DE!
MCQ. A: -5 B: -4 C: -3 D: -2 E: -1
Subexercise (B). Write down the linear approximation of the DE around the smallestyf ix fixedpoint as dtd∆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 sometc>0, then answer±∞.)
MCQ. A:−2 B:∞ C: −1 D:−∞
Subexercise (D). With the given initial condition for y(0), how much is limt→−∞y(t) ? (If limt→tcy(t) =±∞
for sometc<0, then answer±∞.) MCQ. A:−2 B:∞ C: −1 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 -2.5
-2.0 -1.5 -1.0 -0.5
0.0 0.2 0.4 0.6 0.8 1.0 -2.5
-2.0 -1.5 -1.0 -0.5
0.0 0.2 0.4 0.6 0.8 1.0 -2.5
-2.0 -1.5 -1.0 -0.5
0.0 0.2 0.4 0.6 0.8 1.0 -2.5
-2.0 -1.5 -1.0 -0.5
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Exercise (2). Time-independent DE, two dimensions. Qualitative behaviour, linearization.
Let
d dt~y=
f1 f2
=
−(y1+ 2) (y2−1) (y1+ 1) (y2+ 2)
.
Subexercise (A). Find the sum of the coordinates of the smallest~yf ix fixed point!
MCQ. A: -5 B: -4 C: -3 D: -2 E: -1
Subexercise (B). Write down the linear approximation of the DE around the smallest ~yf ix as dtd∆~y = A∆~y ! How much is the sum of the elements ofA ?
MCQ. A: 1 B: 2 C: 3 D: 4 E: 5
Exercise (3). Hom.Lin. DE. Radioactiv decay: I →II →. . . Let
A=
−5 0 5 −6
, d
dt~y =A~y, ~y(0) = 4
2
.
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 (~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: 95 B: 115 C: 135 D: 145 E: 165
Subexercise (C). The solution of the DE can be written as
~ y(t) =
2
X
i=1
Cieλit~vi.
How much isC1, if~y(0) satisfies the given initial condition?
MCQ. A:−20 B: −18 C: −16 D:−15 E: −14
Subexercise (D). Compute the matrix exponentialetA ! How much is e1.4·A
21 ? MCQ. A: 0.00305775 B: 0.00343507 C: 0.00385896 D: 0.00433516 E: 0.00487012 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
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(0.512A))13 ?
MCQ. A: 0.116674 B: 0.131072 C: 0.147246 D: 0.165416 E: 0.185829 Subexercise (C). Radioactive decay, I →II →. . ..
d dt~y=
−3 0 6 −3
~ y=A~y.
(Itt~y= (yI, yII)T.) How much is (exp(0.512A))21 ?
MCQ. A: 0.588587 B: 0.661218 C: 0.742813 D: 0.834476 E: 0.93745 2
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) + 2G(t) =δ(t), G(−1) = 0.
How much isG(0.3) ?
MCQ. A: 0.548812 B: 0.616535 C: 0.692615 D: 0.778084 E: 0.8741 Subexercise (E).
G00(t) + 2G(t) =δ(t), G(−1) = 0, G0(−1) = 0.
How much isG(0.3) ?
MCQ. A: 0.182758 B: 0.20531 C: 0.230645 D: 0.259107 E: 0.291081 Exercise (6). Laplace transform
Subexercise (A). Let
y0(t) + 2y(t) = 3t+ 2, y(0) = 6.
How much isY(1.6) ?
MCQ. A: 2.08244 B: 2.33941 C: 2.62809 D: 2.9524 E: 3.31673 Subexercise (B). Let
y0(t) + 2y(t) = 3t+ 2, y(0) = 6.
If
Y(s) =X
k,n
Ak,n
(s−αk)n, Ak,n6= 0, α1> α2 >· · · , then how much isA1,1 ?
MCQ. A:−34 B:−14 C: 14 D: 34 E: 54 Subexercise (C). Let
d dt~y =
−3 0 3 −3
~ y+
1 t
, ~y(0) = 7
3
.
How much isY1(1.6) ?
MCQ. A:{1.65761} B: {1.86216} C: {2.09195}D: {2.35009}E:{2.6401}
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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[0.2,0.7](x) =
∞
X
n=−∞
ˆ
χnen(x), ahol en(x) = einx
√2π on the (−π, π) interval, then how much is|ˆχ2|?
MCQ. A: 0.170254 B: 0.191263 C: 0.214865 D: 0.241379 E: 0.271166 Subexercise (B). Let
φt(t, x) = 3φxx(t, x), φ(t, x) =φ(t, x+ 2π), φ(0, x) =χ[0.2,0.7](x), ha x∈[−π, π].
If
φ(t, x) =
∞
X
n=−∞
cn(t)en(x), Then how much is |c2(0.2)|?
MCQ. A: 0.0154451 B: 0.017351 C: 0.0194921 D: 0.0218974 E: 0.0245996 Exercise (8). Euler-Lagrange equation, numerical methods.
Subexercise (D). (Heun’s method) Let
f(t, y) = (4 + 3t)(3 + 4y) t0 = 4,∆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: 5.49376 B: 6.17169 C: 6.93328 D: 7.78884 E: 8.74999
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