StdNum: 18; DE, Teszt 1, Majus 9.
Exercise (2). Time-independent DE, one dimension. Qualitative behaviour, linearization.
Let d
dty= (y−4)2(y−2)(y−1), y(0) = 1.5.
Subexercise (A). Find the sum of the fixedpoints of the DE!
MCQ. A: 4 B: 5 C: 6 D: 7 E: 8
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: -10 B: -9 C: -8 D: -7 E: -6
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: ∞ D: 1 E: 4
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:∞ E: 4
Subexercise (E). Plot the t→y(t) solutions of the DE!
MCQ.
0.0 0.2 0.4 0.6
1 2 3 4
0.0 0.2 0.4 0.6
1 2 3 4
0.0 0.2 0.4 0.6
1 2 3 4
0.0 0.2 0.4 0.6
1 2 3 4
Exercise (2). Time-independent DE, one dimension. Qualitative behaviour, linearization.
Let d
dty= (y−4)2(y−2)(y−1), y(0) = 1.5.
Subexercise (A). Find the sum of the fixedpoints of the DE!
MCQ. A: 4 B: 5 C: 6 D: 7 E: 8
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: -10 B: -9 C: -8 D: -7 E: -6
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: ∞ D: 1 E: 4
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:∞ E: 4
Subexercise (E). Plot the t→y(t) solutions of the DE!
MCQ.
0.0 0.2 0.4 0.6
1 2 3 4
0.0 0.2 0.4 0.6
1 2 3 4
0.0 0.2 0.4 0.6
1 2 3 4
0.0 0.2 0.4 0.6
1 2 3 4
Exercise (3). Time-independent DE, two dimensions. Qualitative behaviour, linearization.
Let
d dt~y=
f1 f2
=
−(y1+ 4)y2 1−y1
.
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: -9 B: -8 C: -7 D: -6 E: -5
Exercise (4). Hom.Lin. DE. Radioactiv decay: I →II →. . . Let
A=
−5 0 5 −9
, d
dt~y =A~y, ~y(0) = 4
3
.
Subexercise (A). Find the smallest eigenvalue ofA ! MCQ. A: -13 B: -12 C: -11 D: -10 E: -9
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: 125 B: 145 C: 165 D: 175 E: 185
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:−2 B: 0 C: 1 D: 2 E: 3
Subexercise (D). Compute the matrix exponentialetA ! How much is e1.4·A
21 ? MCQ. A: 0.00113564 B: 0.00127578 C: 0.00143321 D: 0.00161006 E: 0.00180875 Exercise (5). Hom.Lin. DE. Overdamped oscillator
Let
¨
y+ay˙+by= 0, A=
0 1
−45 −14
, d
dt~y=A~y, ~y(0) = 4
3
.
Subexercise (A). Find the smallest eigenvalue ofA ! MCQ. A: -13 B: -12 C: -11 D: -10 E: -9
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: −12 C: −10 D:−9 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:−234 B: −214 C: −194 D: −174 E:−154
Subexercise (D). Compute the matrix exponentialetA ! How much is e1.4·A
21 ? MCQ. A: -0.0102207 B: -0.011482 C: -0.0128989 D: -0.0144906 E: -0.0162787
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(1.411A))13 ?
MCQ. A: 0.702135 B: 0.788779 C: 0.886114 D: 0.995461 E: 1.1183 Subexercise (C). Radioactive decay, I →II →. . ..
d dt~y=
−1 0 2 −1
~ y=A~y.
(Itt~y= (yI, yII)T.) How much is (exp(1.411A))21 ?
MCQ. A: 0.485472 B: 0.545379 C: 0.612679 D: 0.688284 E: 0.773218 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: 0.387097 B: 0.434865 C: 0.488527 D: 0.548812 E: 0.616535 Subexercise (E).
G00(t) + 6G(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) = 3t−2, y(0) = 3.
How much isY(1.5) ?
MCQ. A: 0.410841 B: 0.461538 C: 0.518492 D: 0.582474 E: 0.654352 Subexercise (B). Let
y0(t) + 5y(t) = 3t−2, y(0) = 3.
If
Y(s) =X
k,n
Ak,n
(s−αk)n, Ak,n6= 0, α1> α2 >· · · , then how much isA1,1 ?
MCQ. A:−1825 B: −1725 C: −1625 D: −1325 E:−1125 Subexercise (C). Let
d dt~y =
−5 0 5 −7
~ y+
1 t
, ~y(0) = 7
6
.
How much isY1(1.5) ?
MCQ. A:{0.831936} B: {0.934597}C: {1.04993} D:{1.17949} E:{1.32504}
Exercise (9). Fourier transform, heat equation.
Subexercise (A). Letχ[a,b](x) = 1, if x∈[a, b], otherwise it is zero. If χ[0.5,1.9](x) =
∞
X
n=−∞
ˆ
χnen(x), ahol en(x) = einx
√2π on the (−π, π) interval, then how much is|ˆχ2|?
MCQ. A: 0.393138 B: 0.441651 C: 0.49615 D: 0.557375 E: 0.626156 Subexercise (B). Let
φt(t, x) = 3φxx(t, x), φ(t, x) =φ(t, x+ 2π), φ(0, x) =χ[0.5,1.9](x), ha x∈[−π, π].
If
φ(t, x) =
∞
X
n=−∞
cn(t)en(x), Then how much is |c2(0.2)|?
MCQ. A: 0.0251556 B: 0.0282598 C: 0.031747 D: 0.0356646 E: 0.0400657 Exercise (10). Euler-Lagrange equation, numerical methods.
Subexercise (D). (Heun’s method) Let
f(t, y) = (3 + 2t)(2 + 3y) t0 = 3,∆t= 0.01.
d
dty(t) =f(t, y(t)), y(t0) =y0 = 2.
What is Heun’s prediction for y(t0+ ∆t)-re?
MCQ. A: 2.50865 B: 2.81822 C: 3.16598 D: 3.55667 E: 3.99556
Subexercise (E). (Taylor series) Let
f(t, y) = (3 + 2t)(2 + 3y) t0 = 3, d
dty(t) =f(t, y(t)), y(t0) =y0 = 2,
y(t0+ ∆t) =y0+c1∆t+c2∆t2+c3∆t3+O(∆t4).
How much isc2 ?
MCQ. A: 977 B: 978 C: 980 D: 982 E: 983