StdNum: 3; DE, Teszt 1, Majus 9.
Exercise (1). Time-independent DE, one dimension. Qualitative behaviour, linearization.
Let d
dty=−(y−1)2y2, y(0) = 0.5.
Subexercise (A). Find the sum of the fixedpoints of the DE!
MCQ. A: -2 B: -1 C: 0 D: 1 E: 2
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:∞ B: −∞C: 0 D: 1
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:−∞ B: 0 C:∞ D: 1
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
0.0 0.5 1.0 1.5
0.0 0.2 0.4 0.6 0.8 1.0 -0.5
0.0 0.5 1.0 1.5
0.0 0.2 0.4 0.6 0.8 1.0 -0.5
0.0 0.5 1.0 1.5
0.0 0.2 0.4 0.6 0.8 1.0 -0.5
0.0 0.5 1.0 1.5
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Exercise (2). Time-independent DE, two dimensions. Qualitative behaviour, linearization.
Let
d dt~y=
f1 f2
=
−(y1+ 1)y2
−y1(y2+ 2)
.
Subexercise (A). Find the sum of the coordinates of the smallest~yf ix fixed point!
MCQ. A: -7 B: -6 C: -5 D: -4 E: -3
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: 3 B: 4 C: 5 D: 6 E: 7
Exercise (3). Hom.Lin. DE. Radioactiv decay: I →II →. . . Let
A=
−3 0 3 −4
, d
dt~y =A~y, ~y(0) = 3
5
.
Subexercise (A). Find the smallest eigenvalue ofA ! MCQ. A: -8 B: -7 C: -6 D: -5 E: -4
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: 73 B: 103 C: 113 D: 133 E: 143
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:−7 B:−6 C: −4 D:−2 E: −1
Subexercise (D). Compute the matrix exponentialetA ! How much is e1.4·A
21 ? MCQ. A: 0.0239061 B: 0.0268561 C: 0.0301701 D: 0.0338931 E: 0.0380756
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(0.766A))13 ?
MCQ. A: 0.20693 B: 0.232466 C: 0.261152 D: 0.293378 E: 0.329581 Subexercise (C). Radioactive decay, I →II →. . ..
d dt~y=
−2 0 2 −2
~ y=A~y.
(Itt~y= (yI, yII)T.) How much is (exp(0.766A))21 ?
MCQ. A: 0.33107 B: 0.371924 C: 0.417819 D: 0.469378 E: 0.527299 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) + 4G(t) =δ(t), G(−1) = 0.
How much isG(0.3) ?
MCQ. A: 0.212443 B: 0.238659 C: 0.268109 D: 0.301194 E: 0.338362 Subexercise (E).
G00(t) + 2G(t) =δ(t), G(−1) = 0, G0(−1) = 0.
How much isG(0.3) ?
MCQ. A: 0.259107 B: 0.291081 C: 0.327 D: 0.367352 E: 0.412683 Exercise (6). Laplace transform
Subexercise (A). Let
y0(t) + 3y(t) = 3t+ 2, y(0) = 4.
How much isY(1.1) ?
MCQ. A: 1.42745 B: 1.6036 C: 1.80148 D: 2.02379 E: 2.27352 Subexercise (B). Let
y0(t) + 3y(t) = 3t+ 2, y(0) = 4.
If
Y(s) =X
k,n
Ak,n
(s−αk)n, Ak,n6= 0, α1> α2 >· · · , then how much isA1,1 ?
MCQ. A:−13 B: 13 C: 43 D: 53 E: 73 Subexercise (C). Let
d dt~y =
−2 0 2 −4
~ y+
1 t
, ~y(0) = 5
2
.
How much isY1(1.1) ?
MCQ. A:{1.90616} B: {2.14138} C: {2.40562}D: {2.70248}E:{3.03596}
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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[0.6,1.](x) =
∞
X
n=−∞
ˆ
χnen(x), ahol en(x) = einx
√2π on the (−π, π) interval, then how much is|ˆχ2|?
MCQ. A: 0.0975414 B: 0.109578 C: 0.1231 D: 0.13829 E: 0.155355 Subexercise (B). Let
φt(t, x) = 3φxx(t, x), φ(t, x) =φ(t, x+ 2π), φ(0, x) =χ[0.6,1.](x), ha x∈[−π, π].
If
φ(t, x) =
∞
X
n=−∞
cn(t)en(x), Then how much is |c2(0.2)|?
MCQ. A: 0.00884875 B: 0.00994069 C: 0.0111674 D: 0.0125454 E: 0.0140935 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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