StdNum: 10; DE, Teszt 1, Majus 9.
Exercise (2). Time-independent DE, one dimension. Qualitative behaviour, linearization.
Let d
dty=−(y−2)y2(y+ 2), y(0) = 1.5.
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: 14 B: 15 C: 16 D: 17 E: 18
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: −2 D: 0 E: ∞
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: 2 D:∞ E:−2
Subexercise (E). Plot the t→y(t) solutions of the DE!
MCQ.
-0.2 0.0 0.2 0.4 0.6 -2
-1 0 1 2
-0.2 0.0 0.2 0.4 0.6 -2
-1 0 1 2
-0.2 0.0 0.2 0.4 0.6 -2
-1 0 1 2
-0.2 0.0 0.2 0.4 0.6 -2
-1 0 1 2
Exercise (2). Time-independent DE, one dimension. Qualitative behaviour, linearization.
Let d
dty=−(y−2)y2(y+ 2), y(0) = 1.5.
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: 14 B: 15 C: 16 D: 17 E: 18
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: −2 D: 0 E: ∞
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: 2 D:∞ E:−2
Subexercise (E). Plot the t→y(t) solutions of the DE!
MCQ.
-0.2 0.0 0.2 0.4 0.6 -2
-1 0 1 2
-0.2 0.0 0.2 0.4 0.6 -2
-1 0 1 2
-0.2 0.0 0.2 0.4 0.6 -2
-1 0 1 2
-0.2 0.0 0.2 0.4 0.6 -2
-1 0 1 2
Exercise (3). Time-independent DE, two dimensions. Qualitative behaviour, linearization.
Let
d dt~y =
f1 f2
=
(y1+ 5) (y2+ 2)
−y1−2
.
Subexercise (B). Write down the linear approximation of the DE around the~yf ix as dtd∆~y=A∆~y ! How much is the sum of the elements ofA ?
MCQ. A: -2 B: -1 C: 0 D: 1 E: 2
Exercise (4). Hom.Lin. DE. Radioactiv decay: I →II →. . . Let
A=
−5 0 5 −7
, d
dt~y =A~y, ~y(0) = 3
3
.
Subexercise (A). Find the smallest eigenvalue ofA ! MCQ. A: -9 B: -8 C: -7 D: -6 E: -5
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: 75 B: 85 C: 95 D: 125 E: 145
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:−132 B: −112 C: −92 D:−72 E:−52
Subexercise (D). Compute the matrix exponentialetA ! How much is e1.4·A
21 ? MCQ. A: 0.00134429 B: 0.00151018 C: 0.00169654 D: 0.00190589 E: 0.00214108 Exercise (5). Hom.Lin. DE. Overdamped oscillator
Let
¨
y+ay˙+by= 0, A=
0 1
−35 −12
, d
dt~y=A~y, ~y(0) = 3
3
.
Subexercise (A). Find the smallest eigenvalue ofA ! MCQ. A: -9 B: -8 C: -7 D: -6 E: -5
Subexercise (B). Find the~v1 and~v2 eigenvectors corresponding to the λ1< λ2 eigenvalues. Normalize them by the condition (~vi)1 = 1, i= 1,2.Assemble them into a matrixS = (~v1, ~v2) matrixot. How much is the sum of the elements ofS ?
MCQ. A:−14 B: −13 C: −12 D:−10 E: −8
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:−12 B: −11 C: −9 D:−7 E:−6
Subexercise (D). Compute the matrix exponentialetA ! How much is e1.4·A
21 ? MCQ. A: -0.00941006 B: -0.0105713 C: -0.0118758 D: -0.0133412 E: -0.0149875
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(0.542A))13 ?
MCQ. A: 0.116386 B: 0.130748 C: 0.146882 D: 0.165007 E: 0.185369 Subexercise (C). Radioactive decay, I →II →. . ..
d dt~y=
−3 0 9 −3
~ y=A~y.
(Itt~y= (yI, yII)T.) How much is (exp(0.542A))21 ?
MCQ. A: 0.602478 B: 0.676824 C: 0.760344 D: 0.85417 E: 0.959575 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) + 2G(t) =δ(t), G(−1) = 0.
How much isG(0.3) ?
MCQ. A: 0.434865 B: 0.488527 C: 0.548812 D: 0.616535 E: 0.692615 Subexercise (E).
G00(t) + 4G(t) =δ(t), G(−1) = 0, G0(−1) = 0.
How much isG(0.3) ?
Exercise (8). Laplace transform Subexercise (A). Let
y0(t) + 2y(t) = 3t+ 1, y(0) = 4.
How much isY(1.1) ?
MCQ. A: 1.49642 B: 1.68108 C: 1.88852 D: 2.12156 E: 2.38336 Subexercise (B). Let
y0(t) + 2y(t) = 3t+ 1, 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:−94 B:−74 C: −54 D: −34 E:−14 Subexercise (C). Let
d dt~y =
−7 0 7 −3
~ y+
1 t
, ~y(0) = 3
2
.
How much isY1(1.1) ?
MCQ. A:{0.429592} B: {0.482604}C: {0.542157} D:{0.609059} E:{0.684217}
Exercise (9). Fourier transform, heat equation.
Subexercise (A). Letχ[a,b](x) = 1, if x∈[a, b], otherwise it is zero. If χ[0.2,1.4](x) =
∞
X
n=−∞
ˆ
χnen(x), ahol en(x) = einx
√2π on the (−π, π) interval, then how much is|ˆχ2|?
MCQ. A: 0.294629 B: 0.330986 C: 0.37183 D: 0.417714 E: 0.469259 Subexercise (B). Let
φt(t, x) = 3φxx(t, x), φ(t, x) =φ(t, x+ 2π), φ(0, x) =χ[0.2,1.4](x), ha x∈[−π, π].
If
φ(t, x) =
∞
X
n=−∞
cn(t)en(x), Then how much is |c2(0.2)|?
MCQ. A: 0.0267281 B: 0.0300264 C: 0.0337316 D: 0.0378941 E: 0.0425703 Exercise (10). Euler-Lagrange equation, numerical methods.
Subexercise (D). (Heun’s method) Let
f(t, y) = (4 + 4t)(4 + 4y) t0 = 4,∆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: 8.5519 B: 9.6072 C: 10.7927 D: 12.1246 E: 13.6207
Subexercise (E). (Taylor series) Let
f(t, y) = (4 + 4t)(4 + 4y) t0 = 4, d
dty(t) =f(t, y(t)), y(t0) =y0 = 4,
y(t0+ ∆t) =y0+c1∆t+c2∆t2+c3∆t3+O(∆t4).
How much isc2 ?
MCQ. A: 16040 B: 16042 C: 16043 D: 16044 E: 16045